If two objects are in thermal equilibrium, do they have the same thermal or internal energy?

Yes, they do have the same internal energy.

Explanation:

The thermal balance refers to when there is no heat transfer between the bodies and their surroundings i. e. the bodies and the environment are at the same temperature.

Suppose two bodies of different masses and different materials, each one of them is at a temperature of 25 (° C), which is the same temperature as the temperature of the environment, if these two bodies are close to each other, there is also heat transfer as they are at the same temperature, in the absence of any type of energy that enter or exit in these bodies, the amount of internal energy will be equal in both bodies.

Note: when the internal energy of one of these bodies is increased, heat transfer will happen, always looking for the thermal balance.