From a bit of research, I found one listing for the motor inside that fan that describes it as:
MOTOR FOR COOKER HOOD FABER SMEG K25 P25 SX 3 SPEED COMMON WIRE GENUINE SPARE PART 400000144
If that description is correct, then one of the wires is a common wire, and you pick one of the three others to use as a way to determine which of three speeds is present.
Looking at the wiring on the external listing, my reaction is that it is probably the pink wire that is the common, with the other three being the side that determines the speed. I base this on the pink being connected to one of the coils, the other three being connected to the other, and an internal jumper going from the other end of the pink coil to one end of the 3 wire coil. I would definitely want to confirm this an ohmmeter. If the resistance from the pink to all the other wires is greater than the resistance between the other wire colors, then it is almost certainly the common.
So you probably connect the common up to neutral, and try each of the other three to determine what fan speed is best and connect live to that (or one could potentially use a three speed fan switch (like the ones pretty commonly found in store bought fans) to let you switch between the three speeds. (The motor internals suggest blue is high speed, and based upon ordering white is probably medium, and black low speed, but would be totally unsuprised if that was not the correct order.)
The ground (earth) pin should ideally get connected to the metal covering of the fan as exposed metal should generally be grounded (and there is simply no other logical place to connect it). Unfortunately it does not look like the wiring harness contains any wire for that purpose. If it was grounded when installed in the hood, it was probably via however this module was attached to the rest of the hood.
I'm not comfortable advising you on how to make the connections safely, and have no understanding of the rules over there about DIY projects like these. (While generally legal in the States, property insurance is likely to try to cite such creations as a reason for not paying out if something goes wrong, as DIY constructions would not have been properly tested against the relevant safety standards.)