So, I have a blender game file with multiple levels, each in a separate scene. I want to have on "Next Level" scene with a button to go to the next level. Can I make the button always go to the the next level or do I have to make separate "Next Level" scenes?
Typically keys and buttons have a purpose dependent on the context.
According to your question the context of "next level" is a current level.
Therefore you need to determine the current level to find out what operation the button is supposed to perform.
It does not need to be complicated. But there are quite a lot of different solutions.
Level Scene switching itself
Assumption: the level scene is still running (e.g. as background scene)
on receiving "switch to next scene" let an object of the current level scene switch to the next scene.
you start with a level scene, as there is nothing that loads it otherwise
each level scene implements a switching object
Menu scene switching level
on activating your button send a message "switch to next scene" (see above)
I suggest to keep the level in a property either as string or as number. A number is simple as it allows to be incremented.
The important idea is that you first set the level. In reaction of the level change you add, remove or switch scenes (or do other things).
I hope you see this method makes it easy to even add a "previous level" operation.
When you have a lot of scenes you will need to setup a lot of scene setting too. In that case it might be easier to use python to "calculate" the name of the scene that belongs to a level. But this is another topic.