There are many kinds of energy in physics.
There are only two basic forms of energy: kinetic (energy of motion) and potential ( stored energy of position). All "kinds" of energy boil down to being either kinetic or potential.
For example, kinetic energy, potential energy, thermal energy,
electrical energy, Magnetic energy, radiant energy, nuclear energy,
gravitational energy, etc.. are defined physically.
Magnetic, nuclear, and chemical energy are stored potential energy at the molecular or atomic level. Gravitational energy is potential energy associated with the relative position of two or more bodies. Electrical energy may be potential or kinetic energy. Radiant energy, the energy of the motion of electromagnetic waves (or motion of photons), is kinetic energy.
We can 'measure' and 'observe' those energies.
Energy is not a thing in and of itself to be measured or observed. Generally we cannot say this object contains so much energy. But we can measure or observe the consequences of the usage or transfer of energy in terms of work or heat under a given set of circumstances, based on changes in measurable physical properties of substances such as temperature, pressure, volume, etc.. Such measurements or observations show us that energy is always conserved when it changes from one kind or form of energy to another.
...in the case of kinetic and potential energies, the total amount of
energy is measured differently depending on the observer.
That is only true in the case of mechanical kinetic or potential energy. An example is the kinetic and gravitational potential energy of a ball in motion and at a certain elevation where the kinetic and potential energy depend on the frame of reference of the observer. These are the macroscopic (observable) or external kinetic and potential energy of the ball.
On the other hand, the molecular kinetic energy (associated with the temperature of the ball due to molecular motion) and molecular potential energy (associated with intermolecular forces, chemical bonds, etc.) of the ball generally do not depend on the reference frame of the observer. These comprise the internal (microscopic) molecular kinetic and potential energy of the ball.
What I'm curious about is that there are other forms of energy such as
electrical energy, light energy, and thermal energy, but is it also
measured with different values depending on the reference frame?
Thermal energy is an ambiguous term that should be avoided as it sometimes used to describe heat (which is energy transfer due to temperature difference), molecular kinetic energy (which is properly called internal energy), and temperature. The other energy kinds of energy are reference frame independent as previously discussed.
I wonder if some energies can have absolute values independent of the
The kinds of energy that are independent of the observer were discussed above. In general, we do not consider absolute values of energy but rather changes in energy.
Hope this helps.