The most common way would not be, as you say, "de même"/"moi aussi"/"moi de même". In fact, "moi aussi" has a rather vapid feel that has to be avoided. "De même" is not said and even if it is readily understood what one means by that it has something of the dispatched formula you will want to avoid. You might say "Également enchanté", which is somewhat better but not too keen; it sounds too much as if the person saying that should be focusing on the thought of an evaluation of the respective states of mind and that is not the time for that, the moment is consecrated to your expression of the impression the person is making on you. The most common and simple answer is "Enchanté".
The dialogue of introduction to one another of Paul and Amélie is perfect except for one détail: for a short infinitival phrase (te rencontrer), "enfin" is better placed somewhere else.
- Il était heureux d'enfin avoir l'opportunité et le temps de voir Amélie.
- Il était heureux d'avoir enfin l'opportunité et le temps de voir Amélie.
- Il était heureux d'avoir l'opportunité, enfin, de voir Amélie.
— Enchanté de te rencontrer, Paul!
— Enchanté! Ça fait vraiment plaisir de te rencontrer enfin, Amélie!
It goes without saying that there must not have been any contact between the two by means of communications of any sort prior to their first meeting, they must know one another by hearsay only, otherwise the formula "Enchanté" is not proper.