Learn English – word for somebody who cares for his/her child too much

Usually it would be polite not to mention the fact that a child doesn't seem to resemble its parents, since this might imply infidelity on the part of the mother (not good for Susan George in the 1975 film Mandingo). But you will sometimes encounter OED's sense 6a for...

sport - a plant (or part of a plant), animal, etc., which exhibits abnormal or striking variation from the parent type, esp. in form or colour; a spontaneous mutation; a new variety produced in this way.

This noun usage derives from the (now rare) verb sense...

sport (v 8a) - of nature (originally, personified): to ‘amuse herself’ or delight in producing the variety of things in existence, especially abnormal or striking living forms; to produce such forms.

As @ronan comments, facetious "He looks like the postman / milkman / etc." is not uncommon.

There's also throwback (reversion to an earlier ancestral type or character; an example of this). But unless accompanied by a reference to a specific grandparent (or perhaps great grandparent) that the child does resemble, this invariably has negative associations (with the implicit unspecified ancestor being a Neanderthal or some other precursor to homo sapiens).

Kin or family. There's nothing specific to the case.