If you upgrade your soldiers is that the same as making new ones?

Not quite the same no. You will improve the troops that you have but you don't get more troops. Depending on the troops you have and the research that you have done to unlock the options then you can do 2 things:

Promote soldiers, which will upgrade level 3 to level 4 for example. (you can upgrade different levels that you have to the highest level that you have unlocked in the training camps, once you have the research done to do this.)

Enhance soldiers, which will increase their stats slightly over the base for their current level but not as much as increasing their actual level. Enhanced troops heal faster.