My 1991 Civic LX manual transmission (might be the same engine as yours) was
revving up to only 1200 or so, then falling, then revving again, last summer
when sitting at a stoplight and whence, back home, I found air in the EAC
valve. A proper purge fixed this. Would more air in the system cause revs to
5000? Dunno. But if you didn't properly purge the air, then that's the first
thing I would do here.

On the matter "purge the air intake"(?): I think we're not quite on the same
page here.

What I mean is as follows:
After draining and re-filling your radiator cooling system recently, with
the car of course fully cooled,  did you then (1) leave the radiator cap
off; (2) start the car; (3) wait until the radiator FAN came on twice; (4)
added more coolant to keep the radiator neck topped off throughout?

As I mentioned, it takes a long time (30 to 55 minutes) for the car to warm
up enough to make the fan come on.

Please turn quoting back on in your newsreader.

You have an odd problem indeed. This phenomenon could /only/ occur if an
adequate amount of fuel and air were being admitted to the intake, that
amount increasing with the revs, so as to sustain the increase.

I almost think you have a dirty throttle body that is so filthy it's
causing the throttle plate to stick open.

Low coolant results in surging (like with a big air leak), not a steadily
increasing idle that occurs over 4 or 5 seconds.

Questions (please intersperse your answers):

You say it's "recently" been doing this. When is "recently"?

Had the vehicle been subject to heavy rains or driven through deep water?
Had a collision?

What happened to/with the car immediately before it started to rev like
this?

If you were to let it rev, will it eventually reach redline? You have a rev
limiter, so it ought to go no further than abut 6,800 rpm.

What is "purging the air intake"?