The Tridentine Mass gets its name because it was the mass that was codified after the Council of Trent in the late sixteenth century.
What the mass was like before Trent is a little complicated. You see Rome had never imposed one version of the mass on the Latin Rite and there were many, many liturgies used throughout the church. However, the Reformation made it necessary for the Latin Rite to have one primary liturgy so the Council of Trent made the Tridentine Mass the official liturgy, but it also gave permission for the continued use of liturgies that could show they were older than 200 years old.
However, the Tridentine Mass wasn’t a new mass that the Council of Trent just adopted. It’s actually a much older liturgy that really goes back to Gregory the Great in the sixth century. Indeed, the canon is basically exactly the same as when Gregory the Great composed it. However, in the centuries that followed Gregory this form of the mass fell into disuse, typically only being celebrated once a year by the Pope, and the “Gallican” rite became dominant. However, Saint Francis decided he wanted to use this form of the mass for his order as he believed it was the oldest form of the Mass. Thus, the Franciscans through their ubiquity ended up popularizing this form of the mass. At the same time the mass picked up many elements that had been used in the Gallican rite. The Mass of Gregory the Great had very few visual cues so basically all the visual elements of our modern TLM came from the Gallican rite.
By the century preceding Trent the Mass was very, very close to its final form.