I've looked up the etymology of tante, but it just says it comes from Old French ante without saying how it gained the initial "t".
Where did this "t" come from?
The Littré has this hypothesis, ta ante has agglutinated in tante which would have turn into a noun.
It gives some examples in Walloon with which monfré is brother and mononk is uncle.
I found: kimin s' poitt voss monfrér ? = Comment se porte votre frère ?
A similar agglutination exists with monsieur where mon has lost its adjectival status: ton monsieur, mon bon monsieur are possible.