Has anyone noticed this? Whenever you fly 90ª straight up or down the ship starts drifting and it's impossible to level it until you change direction, I wonder if there's a fix.

Which game exactly?
Windows, DOS floppy or DOS CD version?
If DOS have you tried setting "timed" to false in dosbox under the joystick options?
Also, the floppy version of the first X-wing is a bit problematic when it comes to read analog input, they later improved the engine for the CD version.
Also, which joystick do you use?
Which OS?

I use an xboxone controller but happens with any controller or joystick, windows 10 fall creators, the 98 versions and x-wing alliance, I've seen it on youtube videos too, it's weird.

That's weird, it's probably some problem with Windows 10, that OS has a lot of hidden settings that mess with old fashioned joystick interfaces and/or peripherals, so even if you have newer controllers old games that use the old directinput interface tend to act weird.
Unfortunately I decided to stick with 7 exactly for that reason so I can't help you there. :(

It sounds like you are referring to gimbal lock, which affects all X-Wing series games. Not much that you can do other that try to avoid flying at those angles, and when you do, change direction as soon as you can.

Oh, so that's what he meant.
I never considered there to be an identifiable "up" or "down" in X-wing, I thought he was describing some weird joystick stiction or something...

Yes there is a way, open the map it will show you which direction you are going. Ah, thank you this explains everything then.

It has nothing to do with gimball lock, the drift was delibarately programmed in by the developers to prevent a bug from happening where the game doesn't know the orientation of the craft when it points up or down.
It's a shortcomming of the game engine and it's a PIA.