He doesn’t eat a lot like her.
This can mean both she eats a lot and she doesn’t eat a lot, right?
He doesn’t eat a lot unlike her.
Does this have the same problem as the first sentence?
(He doesn’t eat a lot like her) This can mean both she eats a lot and she doesn’t eat a lot, right?
No. As written, this sentence's primary interpretation is that she eats a lot and he doesn't; "eat a lot like her" is a verb phrase.
In order to state that she also does not eat a lot, there should be a comma separating "like her", similar to a slight pause in the spoken sentence. This would clearly make the phrase "like her" stand out as a relative clause, separate from the main clause.
Note that in speech, the pause/comma may be missed, so there is a potential ambiguity in this sentence. Grammatical English sentences can be ambiguous, but it should be avoided if possible.
Better alternatives to this sentence could be
(He doesn’t eat a lot unlike her) Does this have the same problem as the first sentence? (edited question)
This is less likely to cause ambiguity, because "eat a lot unlike her" is not a valid verb phrase. I'd still say a comma is required to separate the main clause "he doesn't eat a lot" from the relative clause "unlike her".
Even better would be "unlike her, he doesn't eat a lot".