gå på museum is a sort of set phrase in Swedish, which means 'going to museums' in general. It doesn't really say whether you're going to one museum or more of them. So while museum is singular, the fact that there's no article makes it into something general. There are other expressions like this too. For instance ha bil literally 'have car'. You can say Jag har en bil to say you have a/one car, but if you just say Jag har bil, you could have one or more cars, the expression is neutral with respect to the number of them.
If you want to state clearly that he likes to go to different museums, you can say that he likes to gå på museer, but in this context it's most likely to guess that this is what is meant anyway, so that it's unnecessary to state it. (not wrong though). And you can translate this sentence into English with either museums, the museum, or a museum, they're all acceptable, but museums is the best translation in this context.