You sing TO someone (as you would talk TO someone) if you are singing ONLY for that ONE person. You sing FOR a reason. You can sing for someone's benefit, honor, or entertainment. You can sing for your dinner or for money.

"to sing for" implies it is like a gift, or it is special for them. "to sing to" implies that you are singing directly to them, face-to-face.

Both make perfect sense.

The difference is subtle. FOR - indicates your reason for singing. TO - basically indicates which direction you sing.