I speak with no experience relative to bicycle landing gear aircraft, but porpoising water borne aircraft is something I have some experience with.
Interestingly, a floatplane and a flying boat have differing tendancies in this regard.
An aircraft will porpoise (mostly on landing) because it can. Pilots have some opportunity to prevent it, or at least "check" it. My experience has been that aircraft whose first contact wheels are very close to the C of G is less likely to porpoise. It' a bit different with a hull.
Consider the basic C-150/152/172. You would probably agree that if you load that aircraft to an aft C of G, and gently push the tail to the ground, it'll darn near stay there on it's own. Thus, in that pitch attitiude, and loading, the C of G is very close to the mainwheels. This is in part because the aircraft is very nose high, and with the aircraft C of G higher than the mainwheels, it moves back as you raise the nose.
So if the ground reacts back to the aircraft on landing, it reacts to the aircraft through the mainwheels (for a "good" landing) and the aircraft reacts back through it's C of G. If one is vertically in line with the other, there is no tendancy for the reaction of the ground and the aircraft to combine to change the pitch attitude - no porpoise.
You can probably relate that if you land that Cessna very nose high, you might drop it on, and bounce a bit, but the pitch attitude remains pretty well what the pilot has set. If you fly that Cessna on at higher speed, it's going to try to porpoise. The plane still wants to fly, and the ground has reacted back up through a line which is aft of the main wheels. It's gonna force the nose down a bit, the pilot will pull up a bit, and the porpoise begins. In my experience, Aztec "F's" are touchy this way.
If you land a taildragger sloppily, the same will happen in reverse. If you bounce on a wheel landing, it's going to pitch the aircraft up, as well as reacting it up, the comination is going to put you back in the air nose up. You'll push the nose down, it will hit again, and the porpoise begins. If you contact the tailwheel first, you're okay for the one moment it takes for the mains to hit, and then the same cycle begins. As long as the tail wheel is on the ground, you do not have pitch control, and you're just along for the ride.
That's why a good "three pointer" is satisfying, if you get it right! A wheel landing can also work very well, as long as you keep the nose low to prevent a return to flight.
So the planes that float... The difference lies in the fact that the point of support is fluid, and thus the reaction of the water up to the plane can change rapidly. Waves, and the fact that water cannot be compressed contribute to the point of support moving relative to the C of G. If it's moving, and the pilot allows a pitch change to result, the porpoise begins. This is worsre than on wheels, as the affect of the water on the hulls(s) is to react back up at a position reltive to the aircraft C of G which can change a little, but quite rapidly.
In the floatplane, you can land fully stalled in the water, get the aft end of the floats in, and still have some pitch control, as the tail is yet further aft, and still able the change the pitch attitude of the plane a little, but enough for th epilot to overcome a porpoise. I very happily land floatplanes fully stalled. Generally you can touch the steps and heels at the same time, and get a pretty gentle landing, certainly with no porpoising.
In a flying boat, a full stall landing is going to give you a real porpoise, and once it starts that way, there is no pilot action to correct it, other than to hold the controls nose up and wait it out. It's going to be too late to put the power on and go around, you're already off the step, so you're not flying back out. Like a taildragger, once a flying boat's aft hull is in the water, the pilot has very little pitch control, and can only hold the attitude nose high to slow it down as fast as possible. Failure to hold the nose high is likely to result in a smashing the nose into the water during a porpoise.
On the other hand, step landing a flying boat, is very likely to result in a great landing, and litle chance of porpoise. It is possible to touch calm water, and hardly feel it, just a tiny tug of the slight drag from contact
with the water.
Taking off either form of water aircraft is where the porpoise can easily happen. You start off pulling to get onto the step, then pushing over on the step. This pull to a push, combined with the moving center of bouyancy can set up the porpoise. The pilot has to react fairly quickly with the un natural motion of pushing the nose down while accelerating during takeoff. Many a Lake Amphibian has been wrecked because it got airborne early off the water during a porpoise, the pilot lowered the nose too far too fast, and smashed back into the water. Lakes got a bad reputation for the high thrust line and difficult to handle on the water, but this was only with pilots who had not received proper training in them. A Lake will generally climb away safely once airborne, if you have the nerve to try it. It's better than the land back hard and porpoise though!