"František and Kateřina like their names."
Translation:František a Kateřina mají rádi svá jména.
First I do not like jejich. That really sounds like it is about names of someone else. You shoud use the reflexive pronoun here. However, that IS accepted here.
The word order was not accepted here. So I added it.
It must be svá as it is accusative plural, svého is singular genitive or masculine singular accusative.