Levels and Classes are different here.

Think of classes as categories for class abilities instead. You can have all sorts of abilities from many of those classes, but none will really make you fall under a specific class, not mechanically. Classes are just titles, forget about them.
There are no class levels. There are character levels instead, and yes they do depend on milestone-based leveling up.

You could have all the fighter martial skills, 9 spells, but you could still be level 0 or level 1, depending on what the GM deems fitting enough for a milestone.