What is an example of a compact manifold $M$ without boundary which does not satisfy the following property:

For every $a\in M$, two pairs $(M\times M, M\times \{a\})$ and $(M\times M, D_{M})$ are homeomorphic pairs, where $D_{M}=\{(a,a)\mid a\in M\}$

Do all spheres satisfy the above property?

Note that every Lie group satisfies the above property, in the stronger geometric version. That is the homeomorphism can be chosen as an isometry of the product metric.