Color shifting in a bipartite graph
The precoloring extension problem is the following:

Input: a number $k$ and a graph $G$ some of whose edges are labeled with labels in $\{1, 2, \ldots, k\}$.

Decision: is it possible to color the edges of $G$ with colors $1, 2, \ldots, k$ such that no adjacent edges share a color and such that each initially labeled edge is colored with the color it is labeled with.

In other words, given a partially colored graph, can you finish coloring it using only $k$ colors?

A specific case of this problem was proven NP-hard in the paper "NP‐completeness of list coloring and precoloring extension on the edges of planar graphs" by Daniel Marx. In particular, the author showed that precoloring extension is NP-hard even when $k = 3$ and $G$ is a planar 3-regular bipartite graph.

As it turns out, this means your problem is NP-hard too. Here's a simple reduction:

Given an instance of the precoloring extension problem consisting of partially colored planar 3-regular bipartite graph $G$ and $k = 3$, we build an instance of your problem:


We set the degree bound/number of colors (which you called $k$) to be $3$
We set $G_1$ to be $G$ with all initially non-colored edges removed
We set $c_1$ to be the partial coloring of $G$ (and since $G_1$ consists of the edges of $G$ that are colored, $c_1$ is a coloring of $G_1$)
We set $G_2$ to be $G$


Notice that $G_1$ and $G_2$ are both bipartite graphs with vertex-degrees at most $3$.

It is possible to color $G_2$ with a coloring $c_2$ that matches $c_1$ everywhere if and only if it is possible to extend the partial coloring $c_1$ on $G_1$ to a full coloring of $G_2 = G$. In other words, finding a coloring for $c_2$ which differs from $c_1$ as little as possible is equivalent to extending the partial coloring of $G$ into a full coloring.

Thus, your problem is NP-hard.

If you need to solve instances efficiently despite the fact that your problem is NP-hard, I recommend using an ILP solver: for each edge-color pair in $G_2$, make a variable that is 1 if the edge has that color and zero otherwise; then use inequality constraints to enforce that the number of colors of each edge is exactly 1 and that the number of edges at each vertex of each color is at most 1; finally, set the objective that the solver tries to maximize to the sum of the variables corresponding with edge-color pairs present in $G_1$.