It is not so easy to explain because English doesn't have a equivalent for "sin" (as f.x. my language does). But let's say-if we translate "sin" like "its own" (to understand the difference between "sin" and "dets/ dens"), so "sin egen" could be translated as "its very own". It means that the crab has its own plate and he doesn't have to share a plate with somebody else. Or- the plate could be even with crab's name on it.
I think it's wrong to translate ‘sin’ with ‘its own’. The difference between ‘sin’ and ‘dens’ &c. is that ‘sin’ refers back to (usually) the subject of the same sentence, whereas ‘dens’ &c. don't imply such a relation. Maybe you could translate it as ‘its own’ if it were used to contrast with another possessive pronoun, but normally you wouldn't.
I'm also nonnative speaker but I can tell you for sure that you're unfortunately mistaken. "Egen" can be used for people,too. It expresses that the thing belongs to smb. or is for smb's personal use. Well, something like that. :-) Exempel: "Jeg har min egen kop" - I have my own cup--and nobody else drinks of it. "Maria har sit egen kontor på arbejde." Mary has her own office on the work--and she doesn't have to share a room with other employees. I hope I could help. I'd ask a native speaker to confirm it, though.