Part of the issue is simply that "dynasty," the traditional English translation for the Chinese term 朝 chao, is imprecise and can cause misunderstandings (like your confusion at how some very short imperial lines can still be considered "dynasties"). The chao is not the lineage itself but the state ruled by the lineage; it has multiple potential definitions, some related to time (as in, the time period in which a regime is in power) and others related to place (as in the physical space where the officials go to attend to the ruler). So when we say "the Qin Dynasty," it sounds like we're referring to the imperial family but we're really referring to the empire.

Qin is considered the successor to Zhou mainly for two reasons. One, most straightforwardly, they conquered and abolished the Zhou state. Two, they conquered the other Warring States and then invented a new title, higher than king, called 皇帝 huangdi (which we conventionally translate as "emperor"). Before the Qin unification the highest title claimed by Chinese rulers was 王 wang, translated as "king"- the Shang and Zhou rulers as well as the rulers of the Warring States all reigned as wang, but once the concept of huangdi was established in the reign of Qin Shihuang (born Ying Zheng), it stuck and remained in use until the fall of the Qing two millennia later. So although the Qin dynasty was short lived, it was immensely influential.

What exactly defined "China" or a "dynasty" naturally also shifted over time. For the Shang and early Zhou, who reigned over only a small region directly but were surrounded by a web of vassals and allies, "China" was a cultural zone made up of many small states and not a united empire. Shang and Zhou "kings" were at the top of the social and religious hierarchy but did not actually directly administer their subjects in the way that later emperors did. But Qin, and the longer-lasting Han that followed, imprinted the idea of a single legitimate regime ruling over the entire Chinese civilization.