Why is this integral divergent?

Daniel Huber suggested in a comment that it's a limitation of Integrate. You can report it to WRI, and they might improve Integrate in a future release. For now, the following is a workaround in V12.2. It is suggested by the OP's observations on what works; namely the suggestion is that if we separate the singular points in the integration intervals, Integrate will evaluate.

Integrate[DiracDelta[1 - y]*DiracDelta[y - 5], {y, 0, 3, 10}]
(*  0  *)

It also works on the general integral, with the assumption that $A<B$; @DanielLichtblau's comment suggests why Integrate does not evaluate the integral without the assumption:

Integrate[
 f[y] DiracDelta[a - y]*DiracDelta[y - b],
 {y, -Infinity, (a + b)/2, Infinity},
 Assumptions -> a < b]
(*  0  *)

Further workarounds

It turns out that splitting the interval is not even needed, just the assumption:

Integrate[
 f[y] DiracDelta[a - y]*DiracDelta[y - b],
 {y, -Infinity, Infinity},
 Assumptions -> a < b]
(*  0  *)

And the specific integral works without splitting if the interval is infinite:

Integrate[
 DiracDelta[1 - y]*DiracDelta[y - 5],
 {y, -Infinity, Infinity}]
(*  0  *)