Is generalized pigeonhole search known to be no harder than PPP?
Please see the comments below by Emil Jeřábek, so I am not that sure anymore that the problem is harder.

No, it is not known but it is harder than PPP :)

Here I focus on the $M=2N+1$ case, so $t=3$, that is, we want $3$ inputs that map to the same output. In Papadimitriou's seminal paper "On the Complexity of the Parity Argument" he defined PPA-3 similarly to PPA as the search problems reducible to the argument "If in a bipartite graph a node has degree not a multiple of 3, then there is at least another such node."

Let me define here PPAD-3 in a way similar to PPAD as the search problems reducible to the argument "If in a balanced, directed bipartite graph $(A,B;E)$ there is a node in $A$ whose outdegree is not $2$ or whose indegree is not $1$, then there is at least another such node in $A$ OR there is a node in $B$ whose outdegree is not $1$ or whose indegree is not $2$."

Just like PPAD$\subset$PPP, you can show that PPAD-3 is a subset of the problem you have defined. Unfortunately this is all the evidence I can give you as (as far as I know) PPA-3 has not been investigated much and no oracle separation results are known for it, though I don't think it would be hard to obtain some. Papadimitriou defining it as a separate class should be enough evidence that there is a good chance that they differ.