How to refer to pets’ names?

Basically the title. I’m wondering how you refer to a pet’s name, since my assumption would be that:
1. “名前” refers to both first and last name, thus using it for pets wouldn’t make much sense
2. That’s a word used for human names

So, how should I ask someone for their pet’s name? Or tell them my pet’s name? As an additional question, are there any cool/useful pet related words or concepts worth knowing? I tried looking these things up but was largely met with “top 100 Japanese pet names” articles