If you restrict to the postulated $m$-dimensional common eigenspace $E$ of $C=xA + yB,$ the fact that $C+\epsilon A$ has the same multiplicity, means that $A$ restriced to $E$ is also a multiple of identity, as is $B$ (by the same argument). Since a small perturbation of a matrix with distinct eigenvalues still has distinct eigenvalues, the situation in the previous sentence must occur for your condition to hold.
If you restrict to the $m$-dimensional common eigenspace $E$ of $C=xA + yB,$ the fact that $C+\epsilon A$ has the same multiplicity, means that $A$ restriced to $E$ is also a multiple of identity, as is $B$ (by the same argument). Since a small perturbation of a matrix with distinct eigenvalues still has distinct eigenvalues, the situation in the previous sentence must occur for your condition to hold.