You’re trying to build a tunnel from the North American plate to the Eurasian plate. There’s no going around it, since you have to cross the boundary somewhere. If you want to go under it, then you’re running your tunnel through the mantle, which is a bit more difficult.
The rift movement is less than one inch per year. Possibly you could account for this with telescoping sections of tunnel that have a sliding seal; two yards of sliding capacity would give close to 100 years of service.
Alternatively, you could deliberately construct your tunnel in a zig-zag fashion and support it on sliding mounts, allowing the whole thing to flex like a coiled spring as the rift separates. The Trans-Alaska Pipeline incorporates both of these features - the zig-zags allow the pipe to cope with thermal expansion/contraction, and together with the sliding mounts in some locations, they can cope with seismic displacement.
The depth would make things difficult. water two miles deep means a bottom pressure of maybe 5,000 psi. Industrial hydraulic systems commonly operate at pressures in this range, so sealing shouldn’t be too difficult - but for structures as large as a tunnel, the walls will need to be extremely thick to prevent implosion.
Short tunnels rely on piston effect for ventilation. Longer tunnels typically have vent shafts with fan-driven flow to assure adequate fresh air. This is going to be a problem for your trans-Atlantic tunnel, since it won’t be feasible to run vent shafts to the surface, and it’s going to be hard to bring fresh air all the way to the middle of the tunnel.
Electric vehicles could minimize the need for air in the tunnel, but how to get the necessary power in there? high-speed trains these days appear to operate on 25-30kV, but they’re never more than a hundred miles or so from a power plant. A wire run of 1800 miles (to the halfway point from each end) at 25kV is going to incur substantial ohmic losses; your train won’t be very efficient. Cross-country transmission lines operate at several hundred kV for efficiency, but you isolate/insulate them by running them high up in the air, which isn’t going to be practical in the tunnel.
Seems to me that it would be possible to build this tunnel, but not remotely desirable. Expensive, time-consuming, potentially unreliable, and slow. High-speed trains are good over distances of a few hundred miles, but since they only run at about 200 MPH these days, a 3600-mile trip would last about 18 hours (versus 6 hours by plane). Some people are afraid of flying, so planes aren’t an option. For them, the choice would be cruise ship on the water for a couple grand, or a tunnel two miles under water for $$15,000 (hey, somebody’s gotta pay for this thing…). And if they’re afraid of flying, how are they going to feel about being in a tunnel crossing a seismic zone, surrounded by 5,000 psi of water?