@HandCandDandG This is an interesting thread. Can you try to adopt them or add to household when they are off your lot? https://answers.ea.com/t5/Bug-Reports/OPEN-CD-Stray-pets-are-not-labeled-as-strays-when-invited-to-l...
March 2020 - last edited March 2020
@HandCandDandG If travelling to a different lot with the stray cat does not fix this problem you can also try this method: With testingcheats enabled you can Shift + Click on any NPC (including stray cats or dogs) and select the "Add to Family" option and they are instantly added to your household (as long as you do not exceed the 8 Sims per household limit).
Good luck, I hope this helps :D