Great question! The long and short of it is that the emergence of states in ancient India is a complicated process, and doesn't often fit into the paradigms that we may be used to in contemporary Europe. Iron Age India, specifically Northern India as far as we know, did indeed have polities that were not monarchies. They called themselves "Gaṇa-Saṃgha", which are translated as "republics" today but were rather more complex. I'll call them "oligarchies".

So here's what we know about them. Indologist Richard Gombrich points out that the Buddha very clearly organised the early Buddhist Saṃgha (roughly "church") around those of older Gaṇa-Saṃghas, in particular, those of the Śākya clan that Buddha belonged to. Monks, for example, had to attend regular councils of the sect, and the older a monk was, the more senior they were. Both these are believed to derive from the oligarchic tradition of a periodic general assembly, where the eldest spoke first.

The distinguished historian Romila Thapar explores this in her seminal "From Lineage Society to State Formation". According to Thapar, both oligarchies and monarchies in North India derived from large clans organised around dominant families with shared descent. Oligarchies tended to be more decentralised and less hierarchical in terms of how clan assets were distributed. Basically, all elite families within one of these polities would have land, workers and cattle of their own. They would meet periodically to vote on decisions for the clan (including taxes, irrigating, and even minting). The rest of the population, who were not male or members of the original aristocracy, had little say in affairs. All this is reminiscent of Athenian democracy, with, however, next to no room for the hoi polloi. However, unlike with Rome and Athens, it is doubtful that Indian oligarchic polities originated from urban cores. Instead, they were often settled by elite pastoral clans, usually to alleviate population pressure, and then grew larger and created new centres of power.

The leader of each family in a Gaṇa-Saṃgha was called a rāja, which today means "king". So Siddhartha Gautama's father was a rāja, making him technically a "prince". But it's doubtful whether this family were as luxurious and well-off as made out to be in later tradition. The origins and growth of that myth belong to a period in South Asian history that is quite distinct to the time when Buddhism emerged, but I won't get into it here.

Now the Śākyas and Mallas were not the only Gaṇa-Saṃghas in North India in Buddha's day. One of the dominant powers of the time was in fact the Vṛjji coalition, which included dozens of smaller oligarchies under its banner. According to Buddhist and Jain sources, it was the Vṛjjis who were the most difficult obstacle in the expansion of Magadhan power, leading to North India's first empire. The classic text on Indian statecraft, the Arthashastra, claims that Gaṇa-Saṃghas were difficult to defeat on the battlefield but could easily be torn apart through internal dissension and strife. Given that there was a considerable variation among these polities, it's doubtful how much information this really gives us, but I thought it was an interesting nugget.

TLDR: Yes, early India did have republics, though they were really oligarchic aristocracies, and these were probably roughly as old as Athens or Rome if not a bit older. Furthermore, they weren't always concentrated on a single city-state.

There is a lot to be said about the governance of Ancient India, more than I can say in a simple and succinct post, especially since a lot of it is shrouded in the mystery of religious texts and philological speculation. Still, most of the secondary sources and histories of India are starting to come around to the idea that India's pre-modern forms tended towards more democratic governance, as opposed to autocratic monarchies. Several histories that discuss the life of the Buddha, two of them being Andrew Skilton's A Concise History of Buddhism and John Keay's India: A History, split the difference and usually hold up Siddhartha Gautama and his family as nobles in democratic polities, similar to a wealthy Roman or Athenian family, that held outsized influence.

How historically attested this is, is unfortunately a bit out of my wheelhouse.

What is attested, and what I can discuss a little bit more about it this bit:

First of all, I had always heard that Siddhartha was a prince, but it seems that while his family was rich and powerful, they weren't exactly nobility, so I feel that was straight up false advertising

The reason you always hear that Siddhartha was a prince is because universally, the sources about the early life of the Buddha explicitly say he was a prince.

That said the earliest full biography of the Buddha that we have is Ashvagosha's Buddhacarita, which, while clearly drawing from earlier canonical sources, itself an interesting document, and serving as the codifying basis for all subsequent biographies of the Buddha, it dates from the first century C.E., roughly five centuries after the events it describes.

Siddhartha Gautama and his family are described as being "Kshatriya" which is one of the four Varna (what is inaccurately but commonly referred to as "castes") of Indian society. The Varna are, in descending hierarchical order: Brahmin, Kshatriya, Vaishya, and Shudra. Finally, usually thought of as a "Fifth Varna" "under," but more accurately completely outside, the four-fold Varna system, are the Dalits, referred to sometimes, but archaically, as "Untouchables."

The Brahmin (not to be mistaken with the Brahman, the traditional Hindu concept for God (to vastly oversimplify)) are the priestly caste. (For this reason, a Bengali professor I knew told me that the term "Hinduism" was a misnomer, since religious authority in "Hinduism" flows from the Brahmins, it is properly termed "Brahmanism.") The warriors and Kings, sometimes joined into a singular "warrior-kings" are the Kshatriya. The Vaishya (of which Gandhi was one) fulfill the skilled labor role, and for that reason are sometimes translated as "craftsmen," "artisans," or "merchants." Finally, the Shudra fill the unskilled labor, referred to alternatively as "laborers," "serfs," or "slaves."

It should be noted at this point that Varna is insanely complex, and nothing short of a PhD dissertation would be able to more than scratch the surface. This four-fold division is itself a vast oversimplification of the phenomenon which was ever-changing with the political, historical, and economic winds. To use just a couple examples, "colorism" has absolutely tinged how the Varna system in India worked, as European colonial overseers preferred lighter-skinned locals to work in the house while darker-skinned locals worked in the fields, regardless of their Varna status. Another example: in parts of India, the Dalits were resigned to sanitation work, and only at night, forced to stay inside, away from the sun, lest members of the higher Varnas lay eyes on them in the daytime. Upon independence, Pakistan offered steady pay to Dalits, (where previously they had to work without much pay) causing a labor and sanitation crisis as a part of their partition with India. Since Islam (and Christianity for that matter) often promoted a comparatively egalitarian society as opposed to the Varnic system, which drew from an appeal to authority, regardless of how specious that authority was, many of India's first converts to the Abrahamic religions were Dalits.

Again, this is a vast oversimplification of a phenomenon that is still very much alive, and constantly changing.

Indeed, how much the Varna influence the people during the time of the Buddha is still very much a topic under debate. What we know is that Ashvagosha refers to the Buddha and his family as Kshatriya, likely because both Sangha tradition, as well as the primary sources that Ashvagosha was privy to (for example, Rahulasutra, itself written down centuries after both Siddhartha and Rahula were both dead) referred to the Buddha as Kshatriya.

There are theological, political, and ideological reasons for the continuation of this motif and the recreation of Kapilavastu (the land the Buddha was born in and his father is said to have ruled) as an absolute monarchy.

There are two theological reasons. First, the Buddha's "Middle Way" approach toward reality was conceived of (so the story goes, as retold by Ashvagosha) after first his birth into profound wealth and comfort, the transformational "Four Sights" or "Four Encounters," and finally Siddhartha's failure to achieve enlightenment as an ascetic. His young life as a member of royalty in profound wealth contrasted with his attempts at enlightenment, living in the forest, often on nothing but rain water for sustenance, are meant to show that the extremes of experience are neither paths towards enlightenment, with hedonism and complete self-abnegation as bookends.

The other reason is that there are many many tales that precede the biography of Siddhartha Gautama's life as the Buddha: referred to collectively as the Jataka Tales, or the Jatakas. The Jatakas, referring to him often as "Buddha-to-be" or "the Bodhisattva" are often tales of morality, showing how the future-Buddha slowly but surely marched towards enlightenment with each of his past lives getting slowly more and more enlightened. With each life, he achieves a new level of self-sacrifice, a new moral pronouncement, and a new level of enlightenment until his eventual attainment of Nirvana. This idea of "good morality leads to a better rebirth" is an idea that is much too massive (much like the history of Varna itself) and would be difficult to separate from the textual sources from oral sources, as there is certainly no shortage of material. However, we can see this concept in play in the Jatakas, which culminate with the Buddha's birth as Siddhartha Gautama, rewarded for all of his previous deeds as the prince of Kapilavastu, only to reject pleasure in all its forms and attempt a life of asceticism, before reaching the last stage of enlightenment and then Nirvana. This is one of the reasons for the Buddha's epithet "Tathagata," literally "The One Who Has Come (And) Gone."

The ideological reasons to the reframing of Kapilavastu* as a Kingdom is that both religious and secular Buddhist communities have tended towards the authoritarian. There are a lot of reasons for this and there is a lot to be said about the democratic and non-authoritarian elements of Buddhism (to name just two: Buddhism promotes that four monks can form their own Sangha, regardless of location, Varna, or situation (though discourages explicitly splitting a Sangha, exiling those who split the Sangha to aeons in hell) and the tulku system in Tibet, which while it was often manipulated in favor of wealthy land-owning families, also had moments where it raised up impoverished peoples into wealth, status, and influence.

*And I should note here that the history of political ideology in India is, like a lot of things in this post, way way too vast a topic to be covered here. And I even feel uncomfortable using the term "Absolute Monarchy" as it's a term that has a relatively specific definition in Early Modern Europe that just does not apply neatly to any of the polities listed in this post. Secondly, I think it's also reasonable to think that the authors retelling the story of the Buddha either (A) did not think so strongly about the political structure of Kapilavastu, (B) assumed it was much like the structure of their own country, or (C) didn't think it mattered that much as the Sangha was historically thought of as "above" politics, regardless of how true that was depending on the circumstances. What mattered to the writers retelling the story of the Buddha was that Siddhartha Gautama was rewarded for all of the deeds of his past lives, and that he was born to extreme luxury and ultimately found no satisfaction in a hedonistic life.