I have gotten the "see vote counts" privilege on Stack Overflow, and I visited a lot of downvoted and upvoted posts on Stack Overflow on the mobile site, but I can't see number of upvotes and downvotes.
Does the mobile site not support it?
No, the mobile site does not support it. Early versions of the app did support it (might have just been development versions), but a lot of phones at the time didn't provide enough accuracy on the touchscreen to make it useful (I asked this same question several years ago).
Perhaps now that a lot of phones have larger screens, this feature may be more feasible...