An interesting variant of maximum matching problem
(As it seems that the connection is not completely obvious, I will write an extended version of the above comments as an answer.)

I will focus on the case of $c = 2$. In that case, the problem can be rephrased as follows:


Let $G = (V,E)$ be a graph. The task is to find a maximum-size $M \subseteq E$ such that the following constraint is satisfied: for each $\{u,v\} \in M$, either $u$ is incident to at most $1$ edge in $M$, or $v$ is incident to at most $1$ edge in $M$, or both.


Equivalently:


The subgraph induced by $M$ has to have the property that all neighbours of a non-leaf node (degree > 1) are leaf nodes (degree = 1).


Equivalently:


The subgraph induced by $M$ consists of node-disjoint stars. 


In what follows I will interpret unmatched nodes (nodes not incident to any edge in $M$) as stars with 0 edges. Hence a feasible solution partitions the set of nodes into node-disjoint stars.

Now if the number of such stars is $k$, then the number of edges in $M$ is exactly $n-k$: there are $n-k$ leaf nodes that are connected to a star center. Therefore maximising the number of edges in $M$ is equivalent to minimising then number of stars.

Now it is straightforward to see that we have a solution with $k$ such stars iff we have a dominating set of size $k$:


Assume that we are given $k$ such stars. Then we can find a dominating set of size $k$: simply take the centers of the stars (in a star with 1 edge you can pick a node arbitrarily).
Assume that we are given a dominating set $D$ with $|D| = k$. Then we can simply connect each node in $V \setminus D$ to one node in $D$. These edges form a family of $k$ stars.


Therefore solving the problem optimally in a certain graph family $F$ is exactly as hard as finding a minimum dominating set in the same graph family $F$. In particular, the problem is NP-hard even in the case of bipartite graphs.

(However, (in)approximability results related to dominating sets cannot be directly applied here. In essence, we have changed the objective function from $\min |D|$ to $\max n-|D|$.)