This will change both the continent and the sea in unusual ways.
1) The ocean floor will create itself anyways.
Bear with me on this one.
If your world has light (solar radiation), it will have a water cycle, because water molecules will be warmed to the point of vaporization. The water cycle drives weathering and erosion, which wears down the land over time.
As sediment slides off the continent on either side of the "coin", it will sink to the lowest point - where both faces of the world meet at the bottom of the sea. Congruent particles will meet each other and remain at the boundary; because the worlds are exact, mirrored copies, matter will run into its counterpart when it reaches the bottom.
Most sediment that reaches the bottom will remain there, because there are no tectonic forces at the boundary that would lift sections of the seafloor back up. Sedimentary rock will form over millions of years, then become metamorphic due to pressure; a muddy floor with rock underneath will inevitably exist.
2) This means the continent will be flatten and shrink over time.
Tectonic and volcanic forces are primarily driven by the convection of rock deep within a planet. Since this world is only a mile thick, it will not have tectonic plate movement, nor will it be able to support any large volcanoes. Therefore, no new landmasses can form - and the continent will continue to erode from the outside-in until the entire world is a shallow sea.
3) The center of the continent will be an inhospitable desert.
Similarly to how super-continents in the past (on Earth) harbored central deserts, your continent will likely be the same. Water is too distant, and most precipitation falls closer to the coast. It's worth noting that this desert will neither be hot nor cold - it should be at about the same temperature as the rest of the world, since there aren't seasons in a world with no axial tilt - but it will be very, very dry.
4) Many sea creatures will be fine.
Plankton and other microorganisms will still be able to thrive, because the light and nutrients they rely on will still be present in this world. Krill and other creatures that feed on these will in turn not be affected, and so on - so you can still maintain a healthy food chain in this world if it's based on those in the shallows on Earth.
5) Say goodbye to deep-sea ecosystems.
This seems most obvious. Creatures will not evolve to be suited for an extreme lack of light, nor will they evolve to feed on the heat and nutrients found in deep-sea vents. As a result, many large Earth creatures that dive deeper may not be able to survive on this world.
6) Two-world creatures may evolve.
Before the seafloor is sufficiently thick, some creatures may take advantage of protection on one side - the world boundary - and develop so that they're attached to their other-side counterparts! While matter cannot be exchanged, and both halves will require energy, decreasing the efficiency of such a system, they may be able to conserve heat energy.
7) There probably won't be any tides.
On Earth, the pull of our moon is responsible for moving large amounts of water, producing tides. It seems like this world will not have a moon, so you may expect to see somewhat smooth sailing.
8) Wind may be wonky.
One of the important wind-driving pressures we have on Earth is the Coriolis Effect. It's caused by the rotation of the Earth, and results in somewhat spiral-like wind patterns.