How much loss is there in traps, really?

W8JI has some measurements, and the worst case loss he found for two traps in a dipole was 1.6 dB. These were coaxial traps, operated at their resonant frequency, which is both the worst way to tune a trap, and the worst kind of trap.

The losses are indeed mostly resistive losses, but not just because there is more metal. Losses are highest at resonance because this is where current in the trap is at a maximum. The current circulating in the trap at resonance is much higher than it would be if there were no trap. Since resistive losses are proportional to the square of current, this makes a big difference.

As such, it's a good idea to move the resonant frequency a little out of the band. This will reduce current and consequently loss in the trap significantly while still providing sufficient impedance for the trap to be effective.

Far away from resonance, the trap is effectively a loading coil. Current isn't very high and so losses in the trap itself are negligible, however the shortening of the antenna reduces radiation resistance and thus makes all the other losses in the antenna system more significant. More traps for more bands means more shortening and less efficiency.

It's hard to say generally just how much loss this ends up being: I'd bet in most situations ground losses are the most significant source of loss, and that can vary quite a bit between local ground conductivity, antenna height (for dipoles), and radials installation (for monopoles).