Different Strokes for Different Folks:
Since you are going with parallel universes, the sky is the limit - which is to say, Apollo and Helios can be the same god in different cultural traditions. How similar are your worlds? Athena can become Minerva, or Kali can become Kalia, Or Callie. Use Actual different names but with the same meaning.
In a sci-fi multiverse of mine, the same child was raised by a Lesbian couple, and in the first universe the child has born of Mother 1, while in the alternate universe Mother 1 was abducted by her family and Mother 2 delivered the transplanted baby. So in one universe the child was named Diana, while in the second, she was Minerva. It was the same genetic person, but with differing names both inspired by Greco-Roman mythology. Diana lived an Idyllic life, while Minerva suffered childhood trauma.
Or use different spellings of the same name. Use tiny one-letter differences, even, and a reader can see even what an ear might not be able to distinguish. So you have John, Jon, Yohan, Juan, Johan Jonas, Johnathan etc.
Otherwise, middle names can be mixed up. Grandpa was more insistent in universe two about them using his name, so John Jacob Schmitt becomes John Terry Schmitt. You can call them John J and John T. Then they could BOTH go by their middle name and still be John Schmitt, but they are called Jacob and Terry for story reasons.
Or keep the names and maintain constant perspective. I had a universe a little like this, and one of the characters was the perspective character. He referred to his alternate selves with nick-names (going with the John theme, local John for the one from the current universe, bad-John for the one that stole his life and threw him into the parallel universe, stupid John for the one that didn't marry the girl they all loved).