TSP Variant --- Colored Path
In this answer we prove the colored path problem is NP-hard for $c\ge 7$. I don't know whether it is NP-hard for $2\le c\le 6$. Without loss of generality, we prove the NP-hardness for  the Hamiltonian Path problem variant, i.e., asking whether there exists such a colored Hamiltonian path, insteand of the TSP variant.
We use the normal reduction from 3SAT (with $n$ variables and $m$ clauses) to Hamiltonian Path with the following changes:

For each variable, we only have $2m$ nodes: $v_1,v_2,\ldots,v_{2m}$. The $i$th clause corresponds to nodes $v_i,v_{i+1}$ and $v_{2m-i+1},v_{2m-i}$, i.e., if the $i$th clause contains the positive literal of this variable, we have the path $v_i\rightarrow c_i\rightarrow v_{i+1}$, where $c_i$ is the node corresponding to the $i$th clause; and if the $i$th clause contains the negative literal of this variable, we have the path $v_{2m-i+1}\rightarrow c_i\rightarrow v_{2m-i}$. In addition, we add a blank node $b_i$ between $v_i$ and $v_{i+1}$.

For each variable, for each $i$, we give $v_i$ and $v_{2m-i}$, $b_i$ and $b_{2m-i}$ the same color. In addition, we give $c_i, b_i, b_{2m-i}$ the same color if the variable is contained in the $i$th clause.
The following graph is constructed from $(x_1\vee \neg x_2\vee x_3) \wedge(x_2\vee \neg x_3\vee \neg x_4)$. The number above a node represents its color.




We remove the edges to the sink node $t$. Instead, we add paths that pass through the blank nodes that may potentially not be visited due to visits of clause nodes. In the example above, the added edges form the following structure.


In the example above, we ask whether there is a Hamiltonian path with colors 1-2-3-4-5-4-3-2-6-3-7-8-7-3-6-...-11-3-8-14. You can check this reduction indeed works.