This is a great question, and unfortunately you've gotten a completely incorrect answer from RC_23.
The first thing to realize is that the analogy is not quite as close as it seems. The relevant quantity in quantum mechanics is the Hamiltonian, and this is actually not always the same thing as the energy of a system. For example, in a rotating frame, they are different things.
Further breaking the analogy is the fact that although in relativity time is just another coordinate, this is not the case in quantum mechanics. In QM, time is a parameter, and there is no such thing as a time operator or a time observable. (For example, if you were to apply a time operator to an electron in the ground state of the hydrogen atom, there is clearly nothing sensible that could come out as a result, and similarly for a time measurement of this system, which is too simple to act as a clock.)
If we look at quantum mechanics from a modern perspective, then foundationally it really has nothing to do with things like momentum or energy. If you look at one of these formulations, e.g., https://arxiv.org/abs/quant-ph/0101012, you will see that the basic building blocks are more like qubits than particles. It's possible to have rich, interesting quantum systems such as quantum computers for which there is a Hamiltonian and a time parameter, but no such thing as momentum or position.
So foundationally, we just have wavefunctions, and the Hamiltonian tells us how the wavefunction evolves with time. There's nothing about position or momentum that's inherent to QM. However, if you are going to have building blocks such as particles living in space, then you are going to have to obey the correspondence principle, which means that you have to be able to recover the classical equations of motion in the appropriate limit. The way we normally do this is by associating the QM Hamiltonian with the Hamiltonian from classical mechanics. If you look at the Hamiltonian equations of motion, then you'll see that there is the kind of symmetrical treatment of Hamiltonian, time, position, and momentum that is described in your question.
In special relativity, the situation is completely different except that we still have to obey the correspondence principle so that we can recover Newton's laws in the nonrelativistic limit. The energy-momentum/time-position analogy is a fully correct one in SR (unlike in QM), and it's not difficult to see where it comes from. Basically relativity is a geometrical theory, so all of our observables basically need to be scalars or four-vectors. (They could also be higher-rank tensors or pieces thereof, but basically they need to be things that use a consistent inner product and parallel transport in a consistent way.) If a particle's world-line is inertial and connects points in spacetime differing by a displacement $(\Delta t,\Delta x)$, then that's some four-vector.
We could ask about this particle's energy and momentum as separate objects, but that won't work, because energy isn't a relativistic scalar, and momentum isn't a four-vector. If we're going to have a quantity that makes sense relativistically and connects in some way to the nonrelativistic E and p, as required by the correspondence principle, then we end up having to have something like an energy-momentum four-vector. Now if we ask about its energy-momentum four-vector, there is no rule we can use to give this four-vector a direction, unless we make it point in the same direction as the displacement vector. (If they pointed in different directions, then in the particle's rest frame, picking a spatial direction for the nonzero momentum would violate rotational symmetry.)
RC_23 says this can be explained by Noether's theorem. This is wrong. Noether's theorem is a consequence of Hamiltonian mechanics, and only in the special case where there is a symmetry. A system does not even need to have such a symmetry, and yet in such a situation all of these relationships involving conjugate variables still have to hold.
RC_23's answer further compounds the error by trying to make some kind of unclear but erroneous claim that this has to do with nonconservation of energy in relativity.
A consequence of this is that taking our universe as a whole, energy is NOT conserved in some respects. If you performed an experiment 1 sec after the Big Bang, and then did it now at the same location, the results would be different because of the expansion of the universe. One way this is manifest is the cosmic microwave background has lost energy over time. This energy hasn't "gone anywhere;" it's just off the balance sheet.
This has nothing to do with your question, which was about QM and SR, not QM and GR. Energy is locally conserved in GR (as expressed by the zero divergence of the stress-energy tensor), but is not globally conserved. This lack of global conservation has nothing to do with the fact that spacetime lacks some symmetry. For example, energy-momentum is conserved for asymptotically flat spacetimes, even though they lack the relevant symmetries. Noether's theorem simply doesn't work as a tool for this purpose in GR, for technical reasons. The relevant symmetry for GR would be diffeomorphism invariance, but Noether's theorem doesn't provide a conserved quantity relating to this symmetry.