stray animals

I have a question about strays.
So my sim lives in Brindleton Bay and since there are so many strays there and i really like cats, i put the trait on my property that stray cats like to hang there. I even put out lots of food for them, and made them a little "outhouse" with cat beds, toys and scratchers. And of course the stray cats all come. What I didn't realize was that the strays also come into my own house ( i don't really mind that though because then my cat can get friends and have "sleepovers".
But what I noticed, was when they first arrived and roamed around the house it said "strays" on them. Fine, makes sense. But after a few sim hours it dosen't say "strays" on them anymore. Just their name. Just like a owned pet. None of them are added to my household or anything.
Why is this? Don't they count as "strays" since i opened my doors? Do i now own all the stray cats in the neighborhood without realizing it? I'm just really confused and hope someone can explain what the deal is