Set of vertex-disjoint cycles maximizing different colored vertices
It cannot be solved in polynomial time, assuming P$\,\neq\,$NP.

Without worrying about colors (i.e. if every vertex had the same color), it is the MAX SIZE EXCHANGE problem from the Kidney Exchange literature and can be solved in polynomial time with a reduction to the Assignment Problem.

With the introduction of colors to the problem, it is NP-hard (and the corresponding decision problem is NP-complete).  The problem that you described can be called TROPICAL MAX SIZE EXCHANGE (TROPMAX EXCHANGE).  CNFSAT reduces to TROPMAX EXCHANGE.

Create a graph with one vertex for each variable in the CNFSAT expression, with all of these "variable vertices" colored the same way.  Assign each clause in the expression its own color.  From each "variable vertex" create two cycles - label one cycle 'true' and one 'false'.  For each literal in the expression, create a vertex that is the color of its clause and put it on the appropriate 'true' or 'false' cycle of the corresponding variable vertex.  Finally, add additional vertices of some new color so that every 'true' and 'false' cycle has the same number of vertices.

Because we are choosing vertex-disjoint cycles, each 'variable vertex' can only be in its 'true' cycle or 'false' cycle, not both.  There are no cycles in the graph other than those true/false cycles.  Every cycle has the same number of vertices, so an algorithm that first maximizes the number of vertices in the cycles can choose any combination of cycles: either true or false for each variable vertex.  If the secondary criterion is to maximize the number of colors in the cycles, then if all colors can be included the CNFSAT expression can be satisfied.

Highley, Timothy and Le, Hoang. "Tropical Vertex-Disjoint Cycles of a Vertex-Colored Digraph: Barter Exchange with Multiple Items Per Agent"
dmtcs:3186 - Discrete Mathematics &amp; Theoretical Computer Science, July 31, 2018, vol. 20 no. 2
https://dmtcs.episciences.org/4690
https://arxiv.org/abs/1610.05115