I think without realizing it, you've stumbled into the complex web of historical linguistics. In linguistics, languages can usually be organized into families and sub-families. It's a bit like biological taxonomy. Languages share things like common vocabulary, root words, grammatical structures, syntax, etc. which can be traced back to common antecedents. The easiest one to look at today is probably the Romance Languages (Italian, Spanish, French, Romanian, etc) tracing back to Latin. This is a relatively rare example of a large group that can be traced back to a recorded historic language.

There are really two issues here, so I'm answering this in two parts: a first part about historical linguistics and a second part about Nazi Aryanism.

Part 1

More often than not, we have to trace languages back to a reconstructed, theoretical "proto-language." For example, English, German, Icelandic, Norwegian, are all Germanic languages that can be traced back to a Proto-Germanic language that was spoken in northern Germany and southern Scandinavia around 1-500 AD. If we know about migrations of people or the spread of material culture (ie pottery, weapons, etc) from the same origin point as a theoretical proto-language we can often match up the language with the people. You can read more about the technical details of linguistics in this answer from u/keyilan

Latin, the larger Italic family it belongs to, and Proto-Germanic are both part of the much larger Indo-European language family that you seem to have discovered. This covers a huge range of related languages that cover sort of belt from Portugal through almost all of Europe, over the Black Sea, and down around the Caspian into Central Asia, Iran, and India/Pakistan. Many languages spoken in that belt share common vocabulary and structure even if the similarities have faded dramatically over time. By taking all of the common root words, predictable changes in pronunciation, the relatively consistent rate of linguistic change, and physical evidence of migration or cultural spread linguists and historians have traced all of these shared languages back to a so-call Proto-Indo-European (PIE) language which was most likely spoken in southeastern Russia - the area called the Pontic-Caspian Steppe - between 4000-6000 years ago.

At different points, relatively small groups of PIE speakers migrated away from that homeland. As they moved, their culture or political power was strong enough that their language was adopted or forced on the local population in their new homes, and over generations their language became many languages over that huge region. Some went west, and became the Italic, Germanic, Celtic, and Greek languages. Some went as far east as western China and developed the Tocharian language (now extinct). Finally, the ones you're interested in went southeast.

This is called the Indo-Iranian language group. They settled around modern Uzebekistan, Tajikstan, etc, until around 1500 BCE, a group called Indo-Aryan, or sometimes Vedic, went southeast into India. By about 1000 BCE another group went south to Iran. Others, closely related to the Iranian language group, stayed in Central Asia. The Indo-Aryan language meanwhile, was busy developing into Sanskrit and eventually into the languages of northern India today. So while you might not see a lot of direct cultural connections between India and Europe today, Indian languages are very closely related with Iranian languages, which in turn share a lot with Slavic languages and so on.

If you look closely, there are also plenty of cultural similarities in there too, but they tend to be more obvious if you look at ancient history. Afterall, the common ancestor of these connections was a Bronze Age culture. There's a clear connection in things like a horse-riding warrior nobility, and the importance of horses in general. There are lots of connections in religion: a sky-god who his the father and king of the pantheon, Mother Earth, three old women in charge of fate or death, and a god of fire and metallurgy, just to name a few examples.

Often, their names are connect. The usual example is PIE "Dyeus Pater" became "Zeus Pater" and then just Zeus in Greece and "Dju Pater" and then Jupiter in Latin, as well as Dyaus Pita in Sanskrit.

Part 2

Aryan is a word with a particularly fraught history. In the modern world, especially in Europe and the Americas, it is tainted by the German Nazi Party, but Hitler and company didn't invent the word. In fact, it's a very ancient word.

The Indo-Iranians I discussed above called themselves "Arya" or "Aryaman" and it basically meant "a member of the community" or something of that nature. Over time, as they came into contact with other outside groups, it developed into a cultural or ethnic identity. The people who spoke those related languages and practiced their shared religious traditions were "Aryan" or some variation of it for millennia. In India, it mostly fell into obscurity as an archaic word used by the Vedas, ancient Sanskrit religious and mythological literature. In Iran, it remained in use and after a few centuries of linguistic development, it actually became the word "Iran." The name of the country basically just means "Aryan" if you take it far back enough.

Skip forward to the 18th century, and Europeans trained in Enlightenment philosophy and education were having extended contact with India for the first time. Some of these European intellectuals started translating the Vedas and the Avesta the holy text of Zoroastrianism, an ancient Iranian religion. Many of these men were polyglots and in their translation efforts, they began to realize that these ancient Indian and Iranian texts shared a lot of similarities with the languages they already knew. The similarities were closer in the ancient European languages. So from those translation efforts they began to theorize that these languages must have some kind of common ancestor. The Avestan and Sanskrit languages were apparently much older, and identified their composers as "Aryan" and so the name was adopted for the theoretical ancestral language.

By the beginning of the 19th century, "Aryan" was being used much like I use "Indo-European" today. The difference is, they hadn't worked out a good candidate for an Indo-European homeland yet. While we are generally confident in southern Russia today, they guessed everything from India to Armenia, to Germany to Atlantis back then.

European race relations being what they were in the 19th century, there was a strong drive to assume that the Aryans originated somewhere in Europe. This was enhanced by a belief that the Aryans were ancient conquerors who defeated everyone in their path and subjugated them. Academics have dismissed that idea today, but the Aryan Invasion Hypothesis was once very popular. Even after academics moved on, the idea of Germany as the Aryan homeland remained very popular in some German circles because of this idea of a powerful, pure ancestor race.

Germany was in a perfect place to latch on to this idea. Historical linguistics really hit its stride and exploded as a discipline around the same time that German nationalism was first developing in the mid-1800s and the two concepts developed together. So, even though no serious academic supported the idea anymore, Hitler and his Nazis were able to latch on to this idea of an ancestral race of super-conquerors called the Aryans.

The use of the swastika is tied into the same historical process. 19th century academics and casual observers alike quickly noticed that the swastika appeared all over the world, but was most widely used in India, the same place where the word Aryan originated. As a result, the symbol was tied to the idea of an Aryan super-culture in some people's minds. In reality, 4 intersecting right angles is just a really simple design and appears independently in many cultures because of its simplicity.