The big difference is that you are defined by your ancestry so there is little to no mobility.
This seems to be confirmed by the way Americans seem to regard someone's race as a fundamental and immutable characteristic, even while intellectually accepting race is just defined by social convention.
Edit: I might be conflating race with ethnicity.
British people use definitions that are different from those of South Asians, and westerners in general are entirely unaware of caste which is a whole racial hierarchy that affects a lot of the world. Sri Lankans define race as whatever your father was, but Jewishness is inherited from your mother.
It might not be fuzzy within a particular culture at a particular point is time, but it is not consistent between cultures or over time.
Edit: initially omitted because I keep sharing it and might look spammy, but decided to add a link to my blog post on the topic: https://pietersz.co.uk/2023/08/racism-culture-different
> In the US it is very common for someone to identify as half black and half white, or one quarter Irish and three quarters Mexican, and so on.
That is one of the things I find hardest to understand about American definitions. I would just think of someone like that as American (unless they are significantly culturally not American in some way). I just describe myself as mixed, or as from one of the two countries I am a citizen of (I have lived in both for significant chunks of my life and am influenced by both cultures but have little ancestry from either).