The game mostly avoids using pronouns - generally, it uses names - but there are times when using the name would be too repetitive. It's a bit of a sticky wicket in English - we can use 'they' and 'their' in some contexts (and yes, it's perfectly correct English), but in others, it doesn't sound quite right. I think, however, that in most cases, it's perfectly clear that 'they' would be referring to a particular Sim, not a group of Sims.
If I see 'Kate just caught a new fish! Check their inventory!', I would understand 'their' as a pronoun substituted for 'her' or 'his' to avoid identifying Kate as male or female. I think I'd find it less awkward to read than 's/he' or 'her/his', and 'their' is actually more gender-neutral, inclusive of people who think of themselves as neither or both.
At the same time, you've got Sims in many languages, and I agree with kwanzaabot - it would require a lot of editing. It's not just a simple substitution - in English, it would also affect the verb - you can't say 'they is', even if you're referring to one person. 'He is' has to change to 'They are'. In other languages, you have to find a similar convention, and it gets more tricky when adjective endings have to match. (And let's not even get started on inflected languages.) On the other hand, some languages are easier - in French, a possessive pronoun matches the thing, not the possessor.
Probably something to think about for Sims 5 - it's the sort of thing that needs to be done from the start.