The repeated "Recover!" is indeed the formation leader, not an automated warning. It's as regular as it is because pilots, and especially military pilots, are trained to maintain a very strong radio discipline, and also (as Brian Shul notes in Sled Driver, his SR-71 memoir) because as a point of personal and cultural pride everyone wants to sound like Chuck Yeager on the radio. It makes sense, too; things can go wrong very fast in an airplane, and it does no one any good for a pilot to lose their cool in an emergency.

Listen closely, and you'll notice that the formation leader lets that discipline fray a bit in the second and following calls, when some urgency bleeds through. This is because he suspects his trainee is about to die, which is what would indeed have happened if AGCAS hadn't been available.

(You can also tell it's a radio call and not an annunciator by the form of address - the full call is "Two, recover!"; "Two" is what someone else in the formation calls its second member. "One" is the formation leader; in a four-ship formation you'd also have "Three" and "Four", and so on for larger groups.)

edit: corrected some terminology