I'll add another point that often gets overlooked - it's the difference in conditions. A player flies to their national team, maybe to a different part of the world, the climate changes, time zones, field quality, even the food. The body doesn't have time to adapt, then he quickly flies back to his club where everything's different again. It's not just fatigue, it's stress for the whole body, and injuries often come exactly when the immune system and muscles are working in constant rebuild mode.
Plus the medical staff at the national team and the club work differently - different recovery protocols, different warm-up approaches. A player at the national team might get a micro-injury that just goes unnoticed there or gets downplayed, because the doctors are focused on getting him through the tournament. Then he comes back to his club where the load suddenly changes, and that old micro-injury suddenly becomes a serious problem.
Basically, it's not just about the number of matches, but the fact that the body ends up in constant "shock" mode - new conditions, new game system, new doctors, new routine. That's why after international breaks injuries hit in waves.