It is very common in most songs with lyrics for the end of a vocal phrase to rhyme with the end of another vocal phrase. It also very common for solo instruments to play the same phrase twice and have a slightly different ending the second time. Would this be considered the melodic equivalence of a rhyme and is there a term for this?
What you describe sounds almost like the opposite of a rhyme, in that it is the beginning of the phrase that matches instead of the end.
Anyway, that kind of melodic writing is of course very common, and would usually be described as an antecedent and consequent phrase.
In some contexts, that might be considered sequencing -- typically used to describe short motifs that repeat quickly with a different interval at each iteration.