Short version: why does the RPE depend on the bike I'm riding even if the same total amount of work is used?

Longer version: I'm riding up a hill (4-6% grade, total time about 14-15 minutes) with a specific amount of power (let's say 220W average). On one day I use a heavy (steel) bike (10kg), on another day I use a lighter (carbon) bike (7kg); both bikes have the same setup wrt my position on them. Obviously it's faster going uphill with the lighter bike. However, the RPE for the heavier bike is also noticeably higher than for the lighter bike.

Why is that? Is it just psychological?
Some possible reasons I came up with are:
- just because I know that the carbon bike is faster?
- because I'm actually riding a bit faster?
- because the carbon bike provides a "smoother" ride?

I would think more related to position and stressing muscles differently.

The position on both bikes is the same ("both bikes have the same setup wrt my position on them"), so this shouldn't be the cause for the different RPE.

What power measuring device is used on each bike? srm/quarq/powertap?

Polar powermeter (old type) on the steel bike; SRM on the carbon bike.

The power measurements are close to those calculated for the grade, speed, and weight (assuming no wind). Moreover, the average power measured for the steel bike is usually even a bit lower than the one for the carbon bike.

The components (both Campagnolo Chorus) are another difference (besides weight) between the bikes: 10 speed with standard crank for the steel bike, 11 speed with compact crank for the carbon bike. However, the average cadence is about the same (around 100).

Throwing a dart on a board full of random numbers would give as accurate a reading as the Polar. If you're trying to compare RPE on different bikes you need accuracy. The Polar has been proven again and again to be worse than the iBike for estimating power.

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As I wrote: I checked the numbers provided by the Polar powermeter against those that can be calculated. They are within 5% of each. I also validated the data by comparing the differences in the time going up that climb -- these also match the expected values.

There are several other reports that the Polar powermeter is pretty good provided you install it correctly. It doesn't "estimate" power, it measures it. Anyway, I'm sure that this is not the source of the difference in RPEs.
But it seems nobody else has this kind of experience...

You need accurate chain mass and, IIRC, avoid extreme chain angles.

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If it takes longer to climb the same hill, it's more work. (if the power output is the same.)

To give a data point:
I have a short local hill that I rode twice within 3 weeks on the same bike, different setup. 1/2 mile, 10% grade. First time was with my 52/36 chainrings and 12-26 OG-1070 cassette, second time was with 50/34 lighter chainrings and 11-28 OG-1090 Cassette. Time was 3:00 for both, avg HR was 177 for both, estimated average power was 286w first time, 284w second time. Looking at my cadence, with the first setup, I downshifted earlier and my cadence gradually decreased. Second time, I held a higher cadence longer, then had an extra gear to shift down and held that steady for the rest of the way.

Same bike, same watts, same speed, different RPE.

I have a huge problem on the trainer. It seems like maintaining the same wattage (same Powertap wheel, so that variable is covered) is much harder than just going outside and doing it. It's maddening, especially since there is no good explanation. Part of me wants to disbelieve my own RPE, but it happens so often that I have just started to accept a ~20w loss on the trainer.

There you go

Yes, but the RPE is already higher during the climb, not just at the end.

Do you use a huge fan to cool you down when you ride inside? If not, that's the most obvious explanation. AFAIR there are some other discussions about this topic that try to explain the difference.