The issue is their track record. No one believes election campaign promises.
Bloomberg is a centi-billionaire with a track record of loving the free market. His mayoral terms were known for fiscal discipline (vs social welfare spending), hard-on-crime stances (vs. abolish police) and a strong focus on data driven outcomes (vs. intent).
Mamdani is a member of the DSA, which has a track record of being anti-business (Amazon HQ2), pro-regulation (making housing expensive) and pro-union (IMO, NYC unions are a cartel keeping housing & transit more expensive than they need to be).
On paper, they might endorse similar policies, but politicians are judged by their track record and immediate association.