No, Overwatch console players, you're not going mad. There is definitely something wrong with the game's aim assist that's transformed it from hero to mischievous villain. Thankfully, Blizzard is on the case, though you'll need to wait for the fix.
After a few threads popped up on Blizzard's official forums for Overwatch, along with videos highlighting the problem, players finally received a response from game director Jeff Kaplan:
We'll look into this. Thank you for posting.
This was shortly followed up by the game's lead engineer, Tim Ford:
Hi everyone. Thank you all for gathering data on this issue. We've identified the defect and a fix will be available with the next console patch.
If you're not sure how the aim assist bug presents itself, a user by the name of "Nuitch" has a good explanation:
TL; DR: I found an error in the aim assist that basically switches corrections for up and down (I will go into more depth) so when somebody moves up your aim moves down and vice versa. ... What actually is wrong: There are three types of auto aim in Overwatch, strafe lock-in (aim stays on a person when you move sideways), sticky aim, and vertical correction. Vertical corrections (up and down) are switched, so like I had said before when it is supposed to look up it goes down. The implications of this are huge because auto aim becomes auto miss and head shots are nearly impossible.
Of course, nothing quite beats visuals, so here's a clip from one "DrKid" that will annoy you just by watching it:
Aim Assist Broken on Console (Video Evidence) [Battle.net]