Example 1 If the simplest radical expressions $\sqrt[3 x+y]{2 x-y}$ and $\sqrt[y+6]{4 x+y-2}$ are of the same degree, and $y$ is an even number, then the sum of all possible values of $y$ is ( ).
(A) 0
(B) 2
(C) 4
(D) 6
Solution: By the definition of like radicals, we have
$3 x+y=y+6$, which gives $x=2$.
$\because$ The root index $6+y$ is even,
$\therefore$ the radicand $4-y$ and $6+y$ are both non-negative, i.e., $4-y \geqslant 0$, and $6+y \geqslant 0$.
Solving this, we get $-6 \leqslant y \leqslant 4$.
Thus, the even number $y=-6,-4,-2,0,2,4$.
However, when $y=-6$, the root index $y+6$ is zero, making the radical expression meaningless; when $y=-4,-2,0,4$, at least one of the radicals is not in its simplest form.
Therefore, $y=2$. The correct choice is (B).