I have a simple plane object, which rotates perfectly until I rotate along the X-axis, at which point the Z-rotation option rotates around the Y-axis.
Am I missing something, or is this some sort of bug?
I searched for a similar problem, but couldn’t find anything.
Your problem is at the top of your image 
You really should set the gizmo mode to “pivot” and “local” instead of “center” and “global”.
edit Sorry, i just saw that you have rotated the plane by 90° on the x-axis. Your actual problem is a Gimbal-Lock. That’s a problem which appears which eulerangles when you align two axis to point in the same direction. You can “fix” this by parenting the plane object to an empty gameobject which isn’t rotated.
For me, the parenting fix didn’t solve. I don’t really understand why but, for anyone after me with the same problem, here’s how I ‘fixed’ it:
Parenting the object to another one already rotated on X. Then, all the rotations on my object remained normal.