[susan53]

Sorry - I don't quite understand the question. The sentence is fine. I could imagine it being said, for example, in a context where there was contrastive stress on "and" :

A : I don't understand why you always prefer European culture to Asian.
B : But I don't - I like European culture AND Asian.

There are certainly other possible versions of the sentence - eg

I like European culture and Asian culture, too.

which might be more frequently heard. In general when ellipsis happens in clauses after co-ordinators like "and", anything which is the same in the second clause as in the first can be ellipted - ie omitted. The clause thus starts at the first "different" word. Here the full clauses are :

I like European culture, and I like Asian culture too.

"I like" can therefore be ellipted. The first "different " word is "Asian", so everything that follows is included. In writing it would probably be most usual to use this version. But, particularly in speech, it would also be possible to omit other repeated words - as in your example.