my name is supyreor catte: While it's true direct input supports a square range of joystick movement, I have never seen a direct input gamepad with a control stick with anything other than a circular or octagonal range of movement, which just gives you a circle (or octagon) within the square theoretical range of movement. The behaviour in Prince of Persia sounds like badly done direct input support to me.
Obviously I'm aware that flight sticks and the like might have a square range, but I've never seen that on a direct input native gamepad.
And I am the opposite: I never recall seeing an (old) PC DirectInput analog gamepad or joystick/flightick with round(-ish) range of movement. The only "exception" to this rule are the newer Logitech F310/F510/F710 gamepads which have an optional support for DirectInput, but I consider them also as XInput gamepads because that is their primary mode, and overall those gamepads are clearly trying to mimic the XBox360 (XInput) gamepads with their analog triggers, button naming and colors etc.
I have a couple of older DirectInput (I think Logitech and some cheaper unknown brand) PS2-lookalike gamepads, which all have a square range of movement with the analog sticks. I recall wondering in the past how come they have a square range of movement, while e.g. the PS2 gamepads that I also had (used with my PS2 console naturally), had round range of movement.
The analog stick evolution was apparently different on PCs and consoles, on PCs the DirectInput analog gamepads mostly just adopted the same square range of movement that was already present in PC joysticks/flightsticks, while consoles, starting from the first Playstation analog dualshock gamepads, decided to go with a round range of movement.
When XBox360 console gamepads started becoming usable and commonplace also on PC/Windows with the introduction of XInput, then the round area of movement started becoming the norm also on PC, at least on gamepads (frankly I am not sure if PC flightsticks have still retained square range of movement, I'll have to check my new-ish flightstick which I've yet to use, it is still in its box...).
Magmarock: So that's why that happens with sands of time.
It naturally affects other older PC games too which have analog (usually joystick/flightstick) support. I recall trying to play some space game like some Wing Commander or Privateer with my Logitech F310 gamepad, but then I realized I couldn't move the ship diagonally at full speed due to the aforementioned "round vs square" problem.
So I bought a new PC flightstick so that I can play such games with it, if needed. I should try to inspect how easy it is to change the range of the gamepad analog sticks to overcome that problem, so that new XInput gamepads would also be usable with such older PC games.
EDIT: I recall you can even see this behavior if you go to Windows Control Panel/Device Manager, to the utility where you can test all the buttons and analog stick(s) of your gaming controller. When you keep making round movements with the analog stick(s) over and over again, you can't reach the corners in that utility. At least I notice this clearly with my Logitech F310 gamepad (which has a round analog range of movement) in Windows 7.