Me too. No problem to me is a short version of "it was really no problem at all for me to do that for you", and as you say it can also be thought of as "it was nothing" or "think no more about it" etc. It's just a way of letting people know that you were happy to do it, that it wasn't the least bit problematic. It's particularly good at putting people at ease if they were feeling like they asked too much of you in the first place, or were feeling badly about asking the favour.
It is definitely more casual than "you're welcome", and there are times when "you're welcome" is the better choice.