Add crust; remove water
As you point out, the main difference here is the amount of water present on the planet. This is actually even more dramatic than it initially appears - we're not just comparing surface area here, but instead volumes. That is, how does the volume of the crust above sea level (mountains, plains, etc.) compare to the volume of the oceans? A reasonable first-order estimate from Quora using average altitude/depth and surface area finds that the oceans have something like ten times the volume of the crust above sea level, so we're looking at a planet with more like one-tenth the water to get the breakdown shown above.
This actually makes a lot of sense, looking at it a second time. The seas appear to be fairly shallow, aside from that massive hole where the Himalayas were, and the massive newly raised Pacific Plain is going to count for a lot of volume and mass.
Caveat: spirit of the question?
You've gotten a couple comments discussing whether this could occur given continental masses that are still made of basalt and oceanic masses that are made of siliceous materials. To me, the map instead seems to imply that the new continents are indeed siliceous and the ocean crust is still basaltic - and that you're asking about the plate tectonics of such a planet. As in, where are the faults? Are there missing ridges? Can anything here not be explained by the current plate tectonics model? These are the questions I'll be approaching rather than how to make basaltic crust float.
This looks like a supercontinent separation
This might be the Earth-dweller speaking, but this planet looks a lot like the breakup of Pangaea to me. There are several clear cratons present, corresponding to each of the main ocean basins. The largest of these is the Pacific Plain, which would probably be more of a highland/steppe ecosystem due to its 4 kilometer altitude.
Working to the east across the map, we next run into the north and south areas of the American Ocean. This is one place where the supercontinent has a new spreading center forming, likely running down the middle of this ocean and along the Panama Canal. This would be a good example of a back-arc basin. Amusingly, it'd look a lot like the following XKCD proposal:
But the whole thing is underwater, as are (most of) the spreading centers on Earth. Perhaps an overactive transform fault could explain the apparent tearing of the Mid-American Ridge - we get some small offsets on the Mid-Atlantic Ridge in our world, so it's not entirely outside the realm of possiblity.
Arguably, the Andes Trench could also be a consequence of this spreading center. If the East Pacific-Western South American Ocean boundary is active, then the Western South American Ocean plate will be subducting beneath the East Pacific continent, creating a trench. I'd expect mountains to be found a few hundred kilometers into the continent, however, and instead we've got the East Pacific Valley (rather than the East Pacific Rise).
Continuing to the east, it looks like North Atlantic is separating from South Atlantic. The North Atlantic craton is probably headed north, given the lack of Himalaya-style mountain folding in between North and South and instead low land levels that could be a failed (or failing) rift valley, like we see in Africa.
The African Sea is also reasonable - we had a similar setup with the Western Interior Seaway in North America in our own world. Like the WIS, it'd bottom out at 800-900 meters. There'd be plenty of water flowing, so you could either argue that it's an inland sea like the above or separate it into east and west plates and have a mid-ocean ridge running down the middle.
The Mediterranean Archipelago is remarkably similar to the early stages of the accretionary wedges that eventually formed places such as California and Japan. Scattered, oceanic islands are combined together and scraped off a subducting plate onto continental crust. The Mediterranean Archipelago might be the result of some flood basalt events under shallow water, and would eventually be scraped off onto the North Atlantic or Indian plates.
The Indian craton is again a pretty standard, stable plate. Bounded on the left by the rift valley forming between the African and Antarctic oceans and on the right by the Great Australian Sea, it's a large continent that is fairly representative. The Madagascar Sea is annoyingly hard to explain, but is about the same size as our land-locked Caspian Sea, so it should be fine.
The Great Australian Basin probably has another spreading center forming through it. The scattered Indonesian and Philippine seas are a consequence of crustal tearing or hotspot formation, with a main plate boundary running through the Australian Inland Sea, formed similarly to the Western Interior Seaway detailed above. I'd expect it to connect to the Antarctic Ocean sometime soon.
Finally, the Ring of Fire mountains around the Pacific Plain could either be early continent-continent collisions, aggressive accretion by a fast-moving Pacific plate, or orogens resulting from subducted oceanic crust that's undergoing partial melting as it re-enters the mantle. These are essentially a mirror to the volcanoes and faults found around the current Ring of Fire.
Inexplicables
I have no explanation for the Himalayan Ocean Basin. That's just... a weird one. It's gonna be 7.2km deep (much deeper than the current ocean, let alone our new shallower one) and is too circular to be a trench. Maybe an asteroid impact or a George Darwin style moon formation event? Both of those are a stretch to me.
The South Sandwich Mountains are also super weird. I don't see signs of other geologic activity near them, and they're the height of the Himalayas. Similarly, the South Atlantic Plateau just above them is equally difficult to explain - although I personally suspect these have been vertically exaggerrated; that ocean basin is something like 5km deep, less than other places on the planet that haven't been snowcapped.
