Not necessarily, it depends on the game and how it scales to aspect ratios. Some will keep the same vertical FOV, other the same horizontal FOV and a few will adjust arbitrarily.
In this case it really depends on how the game scales aspect ratio and what the calculator is set up to do. Say the game uses vdeg, so the vdeg stays the same between 16:9 and 16:10. If the calculator is also set up to use the vertical FOV, nothing changes. And the same for horizontal.
But if the game uses vdeg and the calculator is set up to use horizontal you will get a difference (and vice-versa).