If you look at sites like Flightradar24.com, you'll see aircraft following the same route, and close enough to allow each other being seen by passengers. Screenshot (look at the scale):
From my personal experience, seeing other aircraft is really common, as already commented and answered, even far from busy airports. I took this picture over Germany, the other aircraft is just a bit higher:
I let who is interested compute the aircraft distance from the size of the static wick. Actually there was a third aircraft in the background, still a common occurrence:
Nearby aircraft can be missed when there is no condensation trail. In general it's easier to spot an aircraft flying at a lower altitude, because Sun's light is reflected by the airframe.
There must be millions of planes and millions of people flying each day
At the time of the question there were about 100,000 commercial aircraft and 9 million passengers flying at a given time in the world. That's not a large number if we consider the size of the planet, but as you said they choose the same routes (for economical reasons). Airways can be very crowded and the absolute separation required for safety purposes is not so large (can be 1,000 ft, 300 m).