You’re saying “takesies” does not rhyme with “backsies”? As a counterexample, what would be a word that would rhyme with one but not the other?
For example, "example" rhymes with "sample" but not with "people" because the stress is on the second-to-last syllable.
But "redact" and "exact" rhyme, despite only agreeing in the last syllable, because that's the last syllable that's stressed.
"Takesies" and "backsies" are both stressed on their first syllables, so they would need to match from that vowel on in order to be perceived as a natural English rhyme, and they don't.