For-loop with else condition runs even though the condition is true

Your indentation is bad on the else clause.

Since it's on the same level as the for loop, the else belongs to the for loop and not the if statement.

Just indent the else clause on the same level as the if like below:

input_namn = input("Ange ett namn: ")
for key in rooms_with:
    if input_namn in rooms_with[key]:
        print (input_namn + "\ndelar rum med \n" + key)
    else: # This is indented correctly
        print ("\nPersonen du angav är inte en del av föreningen\n")

What happens when the else belongs to the for loop ?

Python documentation says:

Loop statements may have an else clause; it is executed when the loop terminates through exhaustion of the iterable (with for) or when the condition becomes false (with while), but not when the loop is terminated by a break statement.

Basically, at the end of the loop the else block is executed except if the loops was stopped by a break statement.

And since there was no break statement in your loop, your else branch was executed.