Question: Would it theoretically be possible for a black hole to be formed solely from very, very intense light? If so, would there be any way to tell the black hole was "made" of light, just by observing it and its surroundings?
It would, but only theoretically. I guess the way you would do it in practice is having several intense light beams cross at one point, where their total energy becomes high enough to form a black hole. However, the required energy density is very high, and therefore this won’t be possible in practice.
Once the black hole has formed, there is no way to tell what it is made of, except whether it has an electric charge. Physicists say that “black hole has no hair”, meaning that all black holes are identical.
I will defer to Arttu’s knowledge on this subject, but is it not also true you could use extremely high frequency light, rather than high intensity, to create a black hole. At high enough energy, two beams of photons would collide just like in the LHC.