My guess would be that every element is different. Going from the basic shell model of the atom to orbitals and transition probabilities there are huge lists for all different transitions. I think there is a green emission for boron where a $11f$-orbital or so is populated and relaxes to give off a green light. And this is so unlikely and weak that you don't even see this on a regular spectrum. So this may be one hint, there are so many possibilities. On the other side if we agreed on one electron, let's say the $1s$-electron, because this is present in all atoms, just exciting that would require different energies for all atoms because the number of protons increase and therefore the positive charge pulling it closer and strong increases as well. It may happen that one transition of element A and a completely different one on element B have the exact same wavelenght, but the spectrum has so many possiblities and there are only very few elements in the PSE for comparison that I doubt that there will be two identical ones. And even if this happens you can strech that spectrum to the UV, X-ray region as well. What is more likely to happen is that the characteristic emission of an element, let's say the one you observe strongest may be similar, like for Lithium, Calcium and Strontium. I doubt that I'd be able to distinguish them by their red color, but then I'd just take a look at the rest of the spectrum.
But I once had a related case. I was preparing a Dysprosium compound and used a sodium based flux. After solving the structure we had the idea that traces of sodium might occupy long anionic-channel in the compound. So we analyzed the elements using energy-dispersive x-ray spectroscopy. And in the spectrum I got were the characteristic Dysprosium emission line and the one from Sodium pretty much on the same spot. So we had to do a wavelenght-dispersive test as well and then I was able to confirm the sodium. This was a case where the strong x-ray emission line of Sodium almost identical to the one of Dysprosium. But in regular spectra you have much more intense emissions so I guess the probability will be really small.