I have the same problem in tons of games, why was walking with the mouse so standard

Two solutions for two different platforms.

1.) Dosbox based games: Novert.com
Simple dos based program that will disable any Y-Axis input from your mouse (no vertical mouse movement)
Download: https://www.doomworld.com/idgames/utils/misc/novert

Add "Novert.com" in your autoexec part of your .conf file

2.) Windows based games: XMBC (XMouse Button Control)
A neat little mouse injector for windows based systems that allows for added mouse functionality.
The program will only effect your mouse when the program is active.
What we want it to is to lock the Y Axis, so windows will no longer accept any vertical mouse input at all.
Download: https://www.highrez.co.uk/downloads/xmousebuttoncontrol.htm

(Settings -> Hotkeys -> Other Hotkeys) Tick "Enable global hotkeys" -> Set "Toggle Y Axis Lock" -> OK -> Apply -> Close

In this case, you want to use XMBC.
Word of warning, this will not "fix" everything mouse control wise with this game.
You will figure out that strafing in older games around this era, was done by applying a strafe modifyer together with a turn key, this means it will also apply to your mouse movements.
TL:DR - Even with this fix, you will not be able to turn & strafe at the same time.

*The issue with turn & strafe at the same time, is something that cannot be corrected from outside of the source-code to the program it self.
Only way it would ever see a change, would be if someone has the source-code to the game or reverse engineered it to such a degree that actual strafe key support could be added (this is possible to do, easy with the source code, quite difficult without and has allready been done in games such as Wolfenstein3D though eighter the SDL source engine or the ECWolf engine)

Enjoy.

Thank you very much :D

Not sure what I'm doing wrong but XMBC doesn't seem to lock the Y-Axis. I did all dyBBelyB said but it only makes my mouse move faster than before both ingame and on my Desktop. Any help to make this work the way it should would be appreciated.

EDIT 06/03/2018: I tried it again using the latest available version of XMBC which is v.2.17 and locking the Y-Axis does work now. To make sure it's active almost the whole time I've bound the lock function to the "W" key, which is also the key I use for moving forward. So whenever I move forward now the Y-Axis movement on my mouse doesn't do anything. Still not a perfect solution but it makes movement in this game a lot easier.

EDIT 08/16/2018: I figured out that there is an easier way to lock the Y-Axis with XMouseButtonControl. All you need to do is add "Lock Mouse Y-Axis (Toggle)" to one of the mouse buttons (1.) under "Layer 1" and hit apply (2.). Optionally you can also save this to a profile (3.).

After that is done start the game (don't close XMBC) and once you are in the main menu hit the button to toggle the Axis lock. If you can't move the cursor up and down with the mouse anymore all is working as intended. Use your arrow keys to start a new game or load a save and enjoy playing with much better controls. I tested this on my Windows 7 x64 system but I guess it will also work with Windows 8 and Windows 10.