Do different oxidation states of the same element have the same emission spectrum?
Interesting question. Keep in mind that the elemental emission spectrum in a flame or plasma and even a discharge does not remember its history in solution or a solid phase.

The punchline is that the emission spectrum is dependent on the elements gas phase chemistry in the flame/plasma/discharge. Prof. Ed has explained you the example of sodium. Let us say we have the following

(a) a block of element sodium
(b) a block of sodium chloride crystal
(c) solution of sodium chloride in water

If you introduce (a), (b) and (c) in the flame, the flame will be colored yellow in each case, which means that the emission is coming from a common emitter. That emitter is a elemental sodium atom excited by high temperature in the gas phase. Thus atomic emission spectrum is a fingerprint of the element.

You may ask that you introduced Na(+) in the flame in the case of b and c. Flames can easily reduce an ion to the elemental state. 

How to see the sodium ion spectrum: As we just said, the emission spectrum is independent of this original state. You can only cause ionization by increasing the temperature. This is the way to see the spectrum of an ion. If we were using a high temperature flame/electrical discharge, we would start seeing Na(+) spectrum along with elemental sodium spectrum.

Coming to your particular example: You introduced copper (I) and copper (II) into a flame and they all colored it beautiful blue-green. The reason is that if the flame temperature is low, compounds cannot fully dissociate into atoms (not enough energy to break the bonds). In such cases, very simple diatomic or triatomic molecules are formed in the flame which emit their characteristic colors. In the case of copper, CuCl is formed in the flame. CuOH may be formed as well. Whether you introduced Cu(I) or Cu(II), as a chloride, it does not remember its solution phase or solid phase history.

If you were indeed using a high temperature flame, you will never ever such a blue green coloration, because this time, the emission is from Cu atoms (in UV). Hope that clarifies your confusion.