Brass fitting seized on a nickel-coated pipe
Both times I have encountered hose faucets with devices seemingly fused onto the hose threads were when somebody (not me) assembled them without a hose washer.
Whether or not this is the case in your situation, you will definitely need to back up your attempts to disconnect the fitting with a substantial wrench on the faucet itself, to prevent twisting the supply pipe or breaking something. Depending on how important this is to you, you may even need to purchase a special wrench due to the recessed configuration of your faucet; maybe something like this:

In the end, you may need to use destructive methods to remove it, which hopefully would preserve the hose faucet but sacrifice the controller. You could use a small extending hacksaw to cut the attachment nut...