Dynastic names were not the same as the names of the ruling families. The Liu family were the ruling clan of the Han, and the Li of the Tang. The early dynasties often took names from places, later ones might use auspicious names (Ming, bright), and sometimes earlier dynasties would be assigned an auspicious meaning later on. (Tang =majesty)

Terms like Songchao 宋朝or Songdai 宋代 (Song/dynasty) would not be used during the Song itself, but would be used to refer to it afterward. During a dynasty it would be referred to with the prefix Da (great) (Da Song, 大宋) and this was part of the official name in Ming and Qing. In everyday use there were terms like benchao (本朝), the current dynasty, or tianchao (天朝) the heavenly (current) dynasty. )

All of the above summarized from

Wilkinson, Endymion. Chinese History: A New Manual, Enlarged Sixth Edition. 6th edition. Cambridge, Massachusetts: Harvard University Asia Center, 2022. Pg 6-9

It took a while but I finally rediscovered an old answer I wrote that covers this: https://www.reddit.com/r/AskHistorians/comments/nr8qo8/why_are_chinese_dynasties_not_named_after_the/h0fhsdx/?context=3

To break it down simply, the term 'dynasty' is not an incorrect translation but it's an incomplete one. The most direct translation of the term chao into English is not 'dynasty' but rather 'court', which encompasses both the imperial house as a family entity and the institution of the imperial state. Indeed, the term chao typically refers not to the imperial house, but rather the state that it ruled. Some historians, especially more recently, have on that basis preferred to use 'Empire' in place of 'Dynasty'.

You ask in your followup to /u/Professional-Rent-62 what the 'nation' was called, and this is perhaps a more complicated question than it first appears. If you actually mean the state, then there was no all-encompassing term that described them all, because it was not perceived that there was a single state that passed between ruling houses, but rather a series of states that were vaguely geographically and ideologically continuous. The Great Qing was the name of the state. If you in fact mean 'nation', then there are a whole load of distinct terms, some ethnic like 'Han' or 'Tang', and some cultural like 'Huaxia', that could be used to refer to certain identity groups, but the evolution of these over time is a huge issue in itself.