In homotopy theory we have the Seifert van-Kampen theorem, which is a clean statement about the fundamental groupoid of a pushout in $\mathsf{Top}$. There is also a 2d version of SvK in R Brown's Nonabelian Algebraic Topology book which again is a statement about preserving pushouts in $\mathsf{Top}$. These theorems do not "leak dimension" in the sense that no knowledge of homotopy objects in other dimensions is needed. This is not to say different dimensions are not related - they are in light of the long exact sequence of homotopy groups.

In homology, the big computational tool seems to be the Mayer Vietoris sequence, which relates the homology of pushouts in $\mathsf{Top}$ to the homology of its components. However, this sequence does "leak dimension" - it does requires knowledge of different-dimensional homology objects to compute the homology of a pushout. There is no SvK theorem for homology groups, even in dimensions 1,2 (as far as I know). On the other hand there is no MV sequence for homotopy.

My question is why.

If we step back, MV is a consequence of the long exact sequence in homology and excision, both of which are available in homotopy. So..

Why is there no MV sequence in homotopy? What (absence of) structure in each theory causes this?

Secondly:

Why is there no SvK theorem for homology in any dimension? What structure is missing in homology that enables this theorem in homotopy?