How do seaplanes do run-up checks without brakes?
I'll limit my answer to single-engine seaplanes as I've never flown a multi-engine seaplane.

Typically there is no need to stay stationary in the water when doing a run-up. Just do it while taxiing to your takeoff path, or you can do it on your takeoff path. If the takeoff path isn't long enough to do the run-up and then continue along the path for the takeoff, you can turn around, taxi back along the takeoff path, and then takeoff.

At typical run-up power settings, if you hold full up elevator, the airplane will simply mush along on the water since you won't have anywhere near enough power to get it up on the step. In other words, you won't be going very fast.

When giving floatplane instruction, one of the first things I tried to impress on students was to not think in terms of the constraints of straight, hard-surface, defined runways, taxiways and the like. There's no need, typically, to taxi in a straight line, make well-defined taxiing turns, and no need to takeoff in a straight line. Once away from the dock or beach, the airplane is going to weather vane unless you actively oppose that, and that weathervaning is going to have a component caused by the current, if any, of the water you're on.

In my experience, most of the time the takeoff path was either dictated directionally (a river) or pretty much wide open (lake or reservoir). For new students in my area, reservoirs were preferred. More often than not, it was a matter of getting a short ways off the dock, do the checks other than the run-up while moving away from the shore at a 90 degree angle more or less, lift the water rudder, bring the stick all the way back, and do the run-up, all the while letting the aircraft weathervane into the wind. Then just takeoff.

I never flew a twin-engined seaplane, but I see no reason why you couldn't bring both engines up to the run-up rpm, do the checks on one and then the other. While checking the mags, carb heat, and props, there would be a little asymmetrical thrust, but not enough that the rudder couldn't handle that.