That’s not how it works, you cannot say anything like
tha mi lèine aig…, it doesn’t make any grammatical sense (I am shirt-ish at…?).
The sentence tha lèine agam means literally a shirt is at me which is a Gaelic idiomatic way of saying I have a shirt – in that sense it is in 1st person, it is me who has the shirt. If you want to change the owner in the sentence, you need to change the agam at me part, eg. tha lèine aige he has a shirt, tha lèine aca they have a shirt, tha lèine aig Mòrag Mòrag has a shirt, etc.
It is explained in the tips and notes for the Pets skill. You can access them in the web browser version of Duolingo (unfortunately currently not available in the mobile app) and on the duome.eu website.