Is it possible to use the phrase: "We are meeting at last" when two people meet for the first time? For example, in a movie one person said: "We meet at last". Why did he use the present simple instead of present continuous? I mean, this was an action, which was happening at the moment of speaking.
Either usage is possible and correct grammatically. There is a subtle difference in emphasis between the progressive and the simple present. The simple present, "we meet", speaks of a an event that is completed as the words are spoken. The progressive, "we are meeting" emphasizes the ongoing nature of the action. By implication, the action of "meeting" may be appropriately regarded as ongoing until concluded by leave taking.
"We meet at last!" is conventional almost to the point of parody while the variant "we are meeting at last" is certainly correct but sounds vaguely exotic.
"Meeting" is something that happens, and then it stops. The meeting in your example is the initial contact. The two people come into contact, which is to say they meet. After that, they've already met. Now they're doing something else, probably having a conversation together.
The speaker in your example could say "So, Doctor Jones, we meet at last and now we are having a conversation." But that wouldn't have the same dramatic flair of a "So, Doctor Jones... We meet at last."