Does this mean "he entertains my daughter"?
No, it means: my daughter is interested in him.
I wonder whether 'he appeals to my daughter' or 'my daughter is interested in him' would be accepted here, and if not, why?
"intéresser quelqu'un" is rather intellectual. "He appeals to my daughter" would be "il plaît à ma fille", which is rather on the emotional side.
Thanks, I see and I see now that above you suggest 'my daughter is interested in him' as a possible alternative. Thank you again!