The concept of entropy does not hold.

Don't you mean the form of the uncertainty relation connecting TIME and energy?

No, I meant just as I wrote. "temperature can only be measured by bringing a system into contact with another system of known temperature. However, energy can only be measured in an isolated system."

This is a classical result independent of quantum mechanics.

Suppose, two monoatomic molecules are trapped in a rather large vessel. After the first elastic collision their kinetic energies would be equalized. From thermodynamic point of view it would mean an equilibrium state with a constant temperature. That temperature is proportional to the kinetic energy of the molecules, so it could be determined using observable values of their velocities, couldn't it? To measure velocities one should add some extra device or he/she could use propostionality between pressure and temperature.

I think along the following lines.

Internal energy is a state function dependent on two macroscopic variables, e.g. U=U(P,V), or U=U(P,T), or U=U(V,T), since the dependence on third variable is fixed by equation of state.

Now, this relation cannot be theoretically established for all kind of systems.

Experimentally, one can always measure the change in internal energy by means of adiabatic work which is path independent.

i.e. One can measure U(final)-U(initial) but one does not have any means to estimate U(initial) because the relationship is not known.

As is known mathematically, entropy is defined as a function of the state of the system, defined up to an arbitrary constant constant. Therefore, you are right, when measuring entropy, each experimenter gets his entropy up to an arbitrary additive constant. :-)

Wouldn't the answer to this question ("can internal energy be measured") depend on what kind of metrology is available (what can be measured in this system) and what kind of knowledge of the system is available?

E.g., suppose, as suggested in the comment by Alexander Kozlov above, that available metrology allows to measure the moments (velocities) and coordinates of each molecule in the system, and the knowledge of the system gives us that the internal degrees of freedom of molecules do not substantially add to the internal energy, also providing the mass of each molecule and the molecule-to-molecule interaction potential. Then, the internal energy of the system can be calculated from the measured momentums and positions of its molecules.

Perhaps, one way to meaningfully formalize the question is to start by postulating that temperature T, pressure P and volume V of any system are measurable using available metrology. Then, one possible formalization of the question is if the internal energy can be measured by measuring T, P or V of the original system or by measuring T, P or V of the result of a certain specially arranged interaction (adiabatic mixing? mixing in a fixed volume?) of the original system with another, specially prepared system. One example of "specially prepared system" (although for measuring T, not internal energy) is a thermometer, which is brought into a contact with a given system and left to arrive to an equilibrium with it, while staying a closed system (no particle exchange with the system being measured). After the equilibrium is achieved, V of the thermometer system is measured and T of the measured system is calculated from it.