How do we know that this is "my sister's" and not "his sister's" or some other variant?
It is assumed in Czech it is my sister. It could also be "your sister" in certain contexts when it is clear we were speaking about "your sister" already. Not accepted here though, "my sister" is the normal meaning here.
How about "I stepped into my sister's car"?
That's somewhat different. I would translate that using vkročil, vstoupil.