I think your first three phrases are appropriate and people would understand. Your final phrase makes more sense as “the squid game is catching on,” which is like saying it is becoming more popular each day. However, I think it is most natural to use the word popular. “The squid game is popular right now.“
“Popular” is a common word in English to describe something well liked by many people.
For the Spider-Man phrase: again, I don’t think the “caught on/catch on” phrase makes much sense. It’s more natural to say “Spider-Man used to be popular but not anymore.”