While the naming conventions used by Japanese rulers have certain Chinese influences, they have historically been distinct nevertheless, so here I'll only discuss the Chinese aspect. To proceed, we need to understand the difference between personal names, posthumous names, temple names, and era names, which I go into depth on in this answer. To restate,

• The personal name is the name an emperor was born with (or switched to later in life). It refers to the person of the emperor, and therefore became taboo during his reign, and would not be used posthumously either. In effect, by convention it is only used to refer to the person of the emperor before he was emperor. Conventions for personal names changed over time; until around the Song period emperors usually had a personal name consisting of the family name (usually one or occasionally two characters) and a one-character given name; from the Song onwards two characters in the given name was more typical. At some point (apologies for not knowing) it also became typical for all the men in a particular generation of a particular lineage to share one character in their given name, creating a surname-generation name-personal name formulation.

• The posthumous name is one of two names by which an emperor could be known by after his death. It also refers to the person of the emperor, but because it was conferred posthumously, it could not be used to refer to a living emperor. The characters in a posthumous name specifically extol virtues of the late person (as another answer in this thread notes, posthumous names were not exclusive to the emperor). Imperial Posthumous names underwent a considerable inflation over time, going from two characters during the early Han to seven during the Tang to a whopping twenty-three for Qing emperors; the typical formulation was to only use the last of these characters, forming a three character phrase, either 'X Y Di' or 'Y Huangdi', with X being the state ('dynasty') name, Y being the character from the posthumous name, and Di/Huangdi being 'Emperor'.

• The temple name is the other posthumously-conferred name for an emperor. Likewise, it refers to the person of the emperor, but being posthumously conferred could not be used for a living ruler. Temple names consistently stayed two characters throughout the ages, and are preferred in the historiography for the Tang period onward. There was a certain degree of formulaic construction to temple names; usually the first emperor of a state was named Taizu or occasionally Gaozu, and often the second was named Taizong; Gaozong was also common for the second or third emperor of a given state. All subsequent emperors were something-zong. The standard formulation, if the state is already clear from context, is just to use the two-character temple name; if the state needs to be specified then the state name is added beforehand.

• The era name or reign name refers to a particular period during an emperor's rule. It therefore does not refer to the emperor himself. Before the Ming, emperors would often divide their reign into multiple eras; the Ming and Qing emperors restricted themselves to just one era per emperor, except in one particular instance where a Ming emperor was deposed for eight years and then came back. Era names were typically two characters, and the construction [era name] Huangdi would be the way to do it. It is worth noting that this means when we see the phrase 'Qianlong Emperor', it doesn't mean 'the emperor named Qianlong', but rather 'the emperor during the Qianlong era'. Hence it is not correct to say 'Emperor Qianlong'.

The thing with all four of these names is it is basically guaranteed that you can avoid repetition within the same ruling house. The classic figure that goes round is that there are some 50,000 characters attested across written Chinese, of which around 20,000 are common enough to warrant inclusion in the more comprehensive dictionaries. That's a lot of characters even if you're just using a one-character given name, and the number of permutations with a two-character formulation is unthinkably vast. Generally, the characters making up names were not reused within the same lineage, at least within several generations' distance, and also never consciously. Granted, only a small fraction of the aforementioned tens of thousands of characters are likely to make it: a parent will hardly name their child, say, 'Stinky'; an emperor would not be likely to be posthumously called 'the Constipated'; nor would he likely refer to his reign as the 'Inauspicious and Disastrous' era. But still, we're talking about a decently wide range of characters to pick from.

So:

• With regard to personal names, emperors are virtually never referred to by their personal names anyway. Plus, in the case of the later [family]-[generation]-[personal] construction, no two emperors from the same family would share a full name anyway owing to the iteration of the generational character.

• With regard to posthumous names, the [state]-[posthumous] formulation means that while the same single-character contracted name might appear across states, it would still be disambiguated. So while there was a Han emperor named Wendi and a Song emperor named Wendi, these could be easily distinguished as Han Wendi and Song Wendi.

• With regard to temple names, again you can use the [state]-[temple] formulation to disambiguate where needed. So for instance Song Taizu can be easily distinguished from Ming Taizu.

• With regard to era names, these were by and large unique formulations each time because these were discrete time periods, not the names of emperors, and consciously chosen not to repeat.

A critical thing to bear in mind is that there was no continuous Chinese Empire with a succession of dynasties, but rather a variety of states, where at various points one at a given time managed to seize control of enough territory to firmly stake a claim to long-term legitimacy as a state in its own right. What that means is that you don't have say 'Taizong I of China' and later down the line 'Taizong II of China'. Emperor Taizong of the Song ruled over a distinct state from, say, Emperor Taizong of the Ming. 'Song Taizong' and 'Ming Taizong' aren't just ways of distinguishing two separate 'Emperors of China', because that title didn't exist to be passed down across states.

Now, theoretically, if a run of emperors chose to reuse the same name over and over, you could conceivably have something bonkers like 'Wen I/II/III of the Qing', but, well, that didn't happen because there are enough characters in the dictionary that you can just find whatever other virtue you want to be associated with and run with that.

This is a tangent to your question, but the modern Thai tradition of regnal numbers is a westernisation. In Thailand itself, the Kings are rarely referred to as Rama (or indeed, in his Thai form, Ram), it's more common to refer to him by a shortened proportion of his regnal name (Vajiralongkorn, as his full name is Phrabat Somdet Phra Paramendra Ramadhibodi Srisinra Maha Vajiralongkorn Mahisara Bhumibol Rajavarangkura Kitisirisumburna Adulyadej Sayamindradhipeshra Rajavarodom Borommanat Pobitra Phra Vajira Klao Chao Yu Hua) or more informally as "Reign No. 10 "(counting from the founding of Bangkok and the establishment of the current dynasty), rather than as Rama X.

While there is a Thai and Southeast Asian tradition (e.g. the Khmer Empire and their multiple Jayavaramans) of regnal names with numbers stretching back centuries (e.g. during the Ayutthaya period, there were mutiple kings sharing names like Sanphet or Ramathibodi), most of these kings also had other regnal names which they are more commonly known by; for example Ramathibodi III is better known as Narai the Great.

This continues today with the current Thai tradition of corwning Kings as "Rama X" etc. this was done by Rama I because he realised that Thai names are long and difficult for foreigners to pronounce, so all the kings in his dynasty have adopted "English" regnal names with Rama [number] for use by foreigners. For example, Rama V's name is Chulalongkorn, he'd be commonly be referred to as as Chualongkorn, or as "Reign No. 5" in Thailand, and the Thai university named after him is called Chulalongkorn University, not Rama V university, or the Mongkutkhlao Hospital, named after a part of Rama VI's regnal name (though, confusingly, not the bit of the regnal name he is commonly referred to by). A more recent example is a dam named after Rama IX - Bhumibol Dam, named after his short-form Thai regnal name.

The only major exception to the Thai use of regnal names is for roads and associated infrustructure projects, which are named as Rama I, Rama II etc. roads. or Rama VIII Bridge, or things named after these roads, e.g. the Honda Rama IX dealership is named that way because it's on Rama IX road, not after the king directly!

I don't have much to add to u/EnclavedMicrostate's excellent answer, but I do want to highlight that during earlier periods of the Chinese history, it was not uncommon to see monarchs adopting elements similar to "regnal number" into their array official titles.

Chinese language has many counter systems. One of which is the 天干 or "Heavenly Branch" counting system derived from the traditional sexagenary cycle date keeping system. Its a very ancient counter system dates back to the Oracle bone script during Shang dynasty. Their use is not limited to traditional calendar. They are still commonly used set-10 counter system in Chinese for both formal and vernacular contexts, basically using the following characters: 1.甲(jiǎ)、2.乙(yǐ)、3.丙(bǐng)、4.丁(dīng)、5.戊(wù)、6.己(jǐ)、7.庚(gēng)、8.辛(xīn)、9.壬(rén)、10.癸(guǐ). For example, 被害人乙 means "victim #2" as "被害人(victim)" + "乙 (#2)," that's the gist of it.

Now please take a look at this table of kings of the Shang dynasty. Focus on the third and fourth columns from the left, which their titles according to traditional historical writings and their temple titles ( 廟號) appear on contemporaneus oracle bone texts respectively. Notice that the vast majority of Shang kings have titles ending in a "Heavenly Branch" counter character. Notably, ALL titles of Shang kings that appear in oracle bone writings end with a Heavenly Branch counter character. As the Rite of Shang did not survive (Confucius actually mentioned in his Analect that he encountered difficulties trying to recover information on the Shang rite), is it still unclear what those counter characters in Shang rulers titles represent. It is true that Zhou and all subsequent dynasties did NOT adopt Heavenly Branch counters in their regnal titles.

Qin dynasty is another time period where "regnal numbers" are formally adopted. Here I will just quote the first Emperor Shihuangdi's edict as recorded in Sima Qian's Shiji :

朕聞太古有號毋謚 (I [Emperor] have learned that in High Ancient times regents only had temple titles but no posthumous title) 中古有號死而以行為謐 (In the Middle Ancient era, regents ruled by their temple titles while alive, and used posthumous titles after death)。如此則子議父臣議君也 (Thus, the newer regnal title system encouraged regents' sons and ministers to judge the worth of their ruler post mortem.) 甚無謂朕弗取焉 (I therefore do not care to adopt such an irreverent system)。自今已來除謚法 (I hereby declare that as of now the old regnal title system has been abolished)。朕為始皇帝 (I shall rule with the regnal title First Huangdi) 後世以計數二世三世至于萬世傳之無窮 (Accordingly. All my successors shall rule under the title based on their numeric order "The Second," "The Third" continuing to "Ten Thousands" ad infinitum." So, after Huangdi The First (literally what Shihuangdi means), his heir ruled under the title "二世" which means "The Second" , and the third and final regent of the crumbling Qin dynasty ruled as "三世" (The Third). This rather unique regnal number system useful by the first Imperial Chinese dynasty was not adopted by subsequent dynasties.