When I tried to add a multiclass to a character, all of the classes say there is a STR requirement of 13 and won’t let me choose them. 

There are prerequisites to multiclass.

I know, but every class says that the character requires a 13 Strength to multiclass. Like, my character with 8 Strength and 16 Charisma can’t multiclass bard.

Yep, because the prerequisite is taken also from your current class.

So if my character does not meet the prerequisite stats for multiclassing into their starting class, they cannot multiclass at all?

That is correct.

Gotcha. Sorry, I had never noticed that caveat before. Thank you for the help!