My suggestion would be to chase the wiring directly rather than trying to identify all the parts of a system your mechanic did not even identify for you.
If possible, find an electrical diagram for the car online. A Google search for 2001 volvo v70 wiring diagram turned up many images of wiring diagrams that should be very helpful.
To trace this back, start at the lights themselves. I suspect the bulbs are fine, but best to be thorough and methodical, you never know. With a rod pressed against the seat and the brake pedal to keep it depressed, see if there is voltage at the bulb bases. (You will probably want to protect the seat with a wood 2x4 or something similar to spread the load rather than press it directly into the seat)
If there isn't any voltage present, use the diagram to tell you where to look next. I'm guessing here but I suspect the fuse for this circuit is as close to the battery as they designers could make it, so it's probably the foot switch. The idea is to divide and conquer. The flow of power to the brake lights is a chain, and any break in the chain will cause them to fail. You are looking for that break. The diagram is your map to the entire chain, from the battery to the brake lights.
If perhaps this box in the trunk is actually in that chain somewhere, the diagram will tell you where power comes into that box, and where it comes out to go to the brake lights. Check the input and output to see if it is the box. If it is, then you will need to identify what it is so you can determine how it works. There is almost certainly a part number molded into the plastic case that can be Googled, or, your local Volvo dealer may be able to identify it for you from the part number.
I hope that helps!