It's a good question, and one that puzzled me for a while as well. However the answer is very simple.
For a massive particle like an electron the total energy is given by:
$$ E^2 = p^2c^2 + m^2 c^4 $$
where $p$ is the momentum and $m$ is the rest mass of the electron. Electrons can obviously have any momentum you want, so the total energy can be any value greater than $mc^2$. The de Broglie wavelength of the electron is $\lambda = h/p$, so the electron can have any wavelength you want.
If we now consider a photon, the key difference is that the rest mass is zero, so the equation for the energy becomes:
$$ E^2 = p^2c^2 $$
Just like the electron, the photon can have any momentum you want, so the total energy can be any value greater than zero. The wavelength of the photon is again $\lambda = h/p$.
So there isn't any difference between the electron and photon except that the non-zero rest mass of the electron means the energy can't be zero. Both electrons and photons can have different energies and wavelengths.
It doesn't matter.
Suppose two electrons approach each other, exchange a photon, and leave with different spins. Are these "the same electrons" as before? This question doesn't have a well-defined answer. You started with some state of the electron quantum field and now have a different one; whether some parts of it are the "same" as before are really up to how you define the word "same". Absolutely nothing within the theory itself cares about this distinction.
When people talk about physics to other people, they use words in order to communicate effectively. If you took a hardline stance where any change whatsoever produced a "different" electron, then it would be very difficult to talk about low-energy physics. For example, you couldn't say that one atom transferred an electron to another, because it wouldn't be the "same" electron anymore. But if you said that electron identity was always persistent, it would be difficult to talk about very high-energy physics, where electrons are freely created and destroyed. So the word "same" may be used differently in different contexts, but it doesn't actually matter. The word is a tool to describe the theory, not the theory itself.
As a general comment: you've asked a lot of questions about how words are used in physics, where you take various quotes from across this site out of context and point out that they use words slightly differently. While I appreciate that you're doing this carefully, it's not effective by itself -- it's better to learn the mathematical theory that these words are about. Mathematics is just another language, but it's a very precise one, and that precision is just what you need when studying something as difficult as quantum mechanics.
Another question, which I think you implied in your (many) questions, is: under what circumstances are excitations related by changes in intrinsic properties called the same particle? Spin up and spin down electrons are related by rotations in physical space. But protons and neutrons can be thought of as excitations of the "nucleon" field, which are related by rotations in "isospin space". That is, a proton is just an "isospin up nucleon" and the neutron is "isospin down", and the two can interconvert by emitting leptons. So why do we give them different names?
Again, at the level of the theory, there's no actual difference. You can package up the proton and neutron fields into a nucleon field, which is as simple as defining $\Psi(x) = (p(x), n(x))$, but the physical content of the theory doesn't change. Whether we think of $\Psi$ as describing one kind of particle or two depends on the context. It may be useful to work in terms of $\Psi$ when doing high-energy hadron physics, but it's useful to work in terms of $p$ and $n$ when doing nuclear physics, where the difference between them is important.
It always comes down to what is useful in the particular problem you're studying, which can be influenced by which symmetries are broken, what perturbations apply, what is approximately conserved by the dynamics, and so on. It's just a name, anyway.
Best Answer
One good piece of evidence that all particles of a given type are identical is the exchange interaction. The exchange symmetry (that one can exchange any two electrons and leave the Hamiltonian unchanged) results in the Pauli exclusion principle for fermions. It also is responsible for all sorts of particle statistics effects (particles following the Fermi-Dirac or Bose-Einstein distributions) depending on whether the particles are fermions or bosons.
If the particles were even slightly non-identical, it would have large, observable effects on things like the allowed energies of the Helium atom.