how can the temperature of a black hole make sense? Is it the temperature of the singularity point? Is it the temperature of the space inside the event horizon (but it can't be, as that space is empty)?
"Temperature" in this case refers to the amount of energy they emit due to hawking radiation.
By comparing to a black-body curve, you can define a temperature for the hole. Obviously it's not a real object with a real temperature -- if it were, I believe the temperature might be infinite -- but this still works for the purposes of deciding whether it'll grow or shrink.
Temperature has several definitions that produce the same number under normal circumstances but may or may not be applicable in extreme circumstances. In this case, I imagine they could be using the thermodynamic definition (dE/dS, the marginal change in energy per marginal change in entropy - I’m not sure if a BH has well-defined entropy) or something to do with the emission curve of space around the black hole. I vaguely recall something about empty space behaving like a blackbody under a gravitational gradient, so maybe they can use that. You could also use amount of Hawking radiation per surface area. It’s possible that some of those produce the same number.
My understanding is it's the temperature of the event horizon: because it doesn't emit any blackbody radiation, other than the tiny amount from Hawking radiation, any heat measurement from it would be approximately 0k.
it is a phenomenon at the event horizon at with radiation is emitted in a way comparable to the radiation produced by every object according to temperature
By comparing to a black-body curve, you can define a temperature for the hole. Obviously it's not a real object with a real temperature -- if it were, I believe the temperature might be infinite -- but this still works for the purposes of deciding whether it'll grow or shrink.