This comes down to the nitty-gritty technicalities of Chinese historiography. The short answer is: Shi Huang Di of Qin was the first emperor of China because he said he was. Someone gotta be the first of something, right? He was so great that he called himself the first emperor, and he was indeed quite great, everyone just accepted it.

The long answer is much longer.

First, we have to understand the title "Emperor of China." There are two technicalities here: what is an "Emperor"; and what is "China"?

The former is somewhat straightforward. The English word "Emperor" with a capitalized E here refers to the Chinese title huangdi (皇帝). There were many great rulers before Qin, and there were even emperors, but they were not called that. If we go reeeally far back to the legendary period of the Three Sovereigns and Five Emperors (c. 3000-2000BC), there were these mythological, demigod rulers called huang and others called di. Now because our boi Ying Zheng (this was his personal name) here was so great for defeating the six warring states and creating a unified, centralized, authoritarian regime, he totally rivaled those mystical god-kings, so he literally justput huang and di together and BAM! You got yourself a huangdi.

So yes, Shi Huang Di was the first huangdi (in fact, his name 始皇帝 literally means "the First Huangdi") because he literally came up with the idea of huangdi. His decedents would be designed the Second Huangdi (二世皇帝), the Third Huangdi, and so on. Now even though the Qin Dyansty was overthrown by the time of Qin Er Shi (秦二世, the Second), the successor Han Dynasty decided the idea of huangdi was actually pretty cool, and so kept it. And thus, from 221BC to 1912 (or 1945), every single ruler who could rule large swaths of China would come to refer themselves as huangdi. It became a title of legitimacy, much like Caesar (or Kaiser, Tsar, etc.) in the West.

So yes. Shi Huang Di is undisputed the first huangdi of China. But he was definitely not the first great ruler of China. As you've said, there were the Shang and the Xia. How are they "dynasties" if they didn't rule all of China... but wait, here comes the second technicality: What even is China?

The modern idea of "China" as a nation-state did not even exist until the 19th century, as the very concept of nation-state was a Western invention. Instead, Chinese people simply referred to their world as... the world. Tianxia (天下), "All Under Heaven." As the Chinese civilization expanded, so did their idea of tianxia. The earlier dynasties of Xia and Shang and Zhou controlled roughly only the Yellow River region, but back then that was quite literally the world to them. Of course, down south in modern-day Sichuan there was the contemporaneous Sanxingdui Culture, but they were so different from the Xia-Shang-Zhou people in every aspect and had effectively no interaction with the Yellow River basin they should well be considered a separate civilization.

Xia, Shang, Zhou, Qin, Han... and all the way to Qing, the ROC, and today's PRC, together constituted a direct, continuous line of succession of what would eventually be known as the Chinese Civilization. There exists direct access of written records in the Chinese language of the ancient regimes such as Xia and Shang, so in traditional Chinese historiography they are regarded as 正统 (I'm not even sure how to translate this, "proper" or "legitimate"?). Even though their political system was in no way similar to that of later Chinese dynasties, even though they probably did not refer to themselves as "dynasties," even though there is little archaeological evidence of their existence at all (for Xia especially), when classical Chinese historians such as Sima Qian were writing the Chinese equivalent of Herodotus' Histories, they nonetheless granted these ancient regimes the title of "dynasty." Perhaps just for ease of periodization, but it stuck, and became canon.