I recently heard someone saying “They have the same towel.” That person wanted to say someone has exactly the same towel as someone else has. To me it sounded a little strange because it sounded like they share and use the same one towel. Isn’t it better to say “they have the same towels”?
They have the same towel is idiomatic and correct (at least in US English). It basically means they have the same kind of towel.
As FumbleFingers says in a comment, if we wanted to say that they jointly have one towel, we would say they share the same towel.
If we said they have the same towels, that would mean that each of them had more than one towel, but all of the towels were the same.