The answer is no. This observation is cataloged at the Group Properties Wiki on pages 1 and 2.
It is possible to have two finite groups $G_1$ and $G_2$ such that the conjugacy class size statistics of $G_1$ are the same as those of $G_2$ (i.e., the two groups have the same number of conjugacy classes of each size) but the Degrees of irreducible representations over $\mathbb{C}$ for $G_1$ are not the same as those of $G_2$...
It is possible to have two finite groups $G_1$ and $G_2$ such that the multiset of Degrees of irreducible representations (over $\mathbb{C}$) of $G_1$ is the same as the multiset of degrees of irreducible representations of $G_2$ (i.e., $G_1$ and $G_2$ have the same number of irreducible representations of each degree) but the conjugacy class size statistics of $G_1$ and $G_2$ are not the same...
Further, the smallest example of the first statement is of size $128$, while for the other statement, examples of groups of size $64$ exist so that these statements are true for $p$-groups as well.
