How does a desktop charging station decide its output voltage?
Since your fan did not negotiate any higher voltage, it was supplied 5V. That's the default for all chargers. It's probably a measurement artifact that you have seen higher voltages.
For your LED project: The QC standard allows to negotiate higher voltages but – as far as I know – does not allow to negotiate higher currents. Since the initial voltage is 5V and since the charger is rated for 5V at 3A, you might be lucky. It will likely deliver 5V at 3A without further negotiation. Just try it. That way you get 15W (5V x 3A).
At 5V, a higher current than 3A is not possible, neither with QC 3 nor with USB PD. At least not officially. If you can and want to work with higher voltages, you can go two routes:

Implement the Quick Charge protocol with a microcontroller. It's not too difficult. See Turning a Quick Charge 3.0 charger into a variable voltage power supply for a good example.

Get a so called trigger board, e.g. this module from AliExpress. It implements the rather complex USB PD protocol. It can be programmed to switch to a certain voltage once plugged in.