I am new to monogame and I never worked with levels before.
Now to my question: is it possible to create a level in a second script/ class, so I can call/ initialize the new level e.g. on button press? Because when I'm initializing the new level in the default script the script is getting pretty messy and large.
Yes, it is possible and surely recommended. For an RPG for example it does make sense to have a separate class or script for each Level / area that defines the behaviour of NPCs or events you may trigger.
This keeps it clean and manageable when the game gets bigger.
I also recommend to build some kind of manager class that "sits" above all your level-classes and handles things like loading/switching levels or distributing player actions. You may also take care about unloading level data when switching between them. It can be quite irritating if some fragments of the previous level are lying around somewhere and cause unexpected behaviour ;)