The answer is yes, but only if there are other optimal solutions than the degenerate one. For example, suppose the primal problem is
$$\max x_1 + x_2$$
$$x_1 \leq 1,$$
$$x_1 + x_2 \leq 1,$$
$$x_1, x_2 \geq 0.$$
The solution $(1,0)$ is optimal and degenerate, but every solution $(a,1-a)$, for $0 \leq a \leq 1$ is also optimal.
The dual is
$$\min y_1 + y_2$$
$$y_1 + y_2 \geq 1,$$
$$y_2 \geq 1,$$
$$y_1, y_2 \geq 0.$$
The dual has the unique (degenerate) optimal solution $(0,1)$. So we do have a situation with a degenerate optimal solution in the primal but a unique dual optimal.
However, if the degenerate optimal solution is unique, then there must be multiple optimal solutions in the dual. The following table is from Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144.
Primal Optimal Solution Dual Optimal Solution
(a) Multiple implies Degenerate
(b) Unique and nondegenerate implies Unique and nondegenerate
(c) Multiple and nondegenerate implies Unique and degenerate
(d) Unique and degenerate implies Multiple