What does the following sentence mean?
I know that it has to do with karaoke but whether it is playing karaoke or karaoke and playing is the problem which I don't know.
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would be a very poorly-written sentence if used in a composition.
In informal speech, however, some people do wind up forming sentences like that. In that sense, it is real-life Japanese.
It would mean:
"I love karaoke and playing (in general)."
In other words, the speaker is listing two things that s/he likes doing. One is specific (karaoke) and the other, non-specific (playing).
We do not say in the first place:
to refer to an action regarding karaoke. We would instead say:
Some might say:
「カラオケで遊ぶ」, but not really 「カラオケと遊ぶ」.