I have 9 pets now. Tried adopting one from the center but it still is saying I already have 10. I'm not sure if this is a glitch but felt it was okay to report just in case.
The pet I sent to the rift is no longer mentioned on the dropdown list nor is on the page with the others, including at the agency.
your slot will never free up. It's not a glitch. If you look your pet up in Search Subeta you will see that for now, the RIFT owns it.