That's a MM-type pickup, so if it's an MMB-4 kit, there won't be another pickup to compare against.
With 4-conductor + ground wiring, the ground wire would normally attach to the metal baseplate of the pickup but here, each half of the pickup looks to have its own baseplate. And from the colour (which would normally be nickel or brass) I am not convinced that this pickup has metal baseplate(s).
So the first thing to do would be to get your multimeter (if you don't have one, then get a basic digital one now - they don't cost a lot and are indispensable when working with guitars) and see if there is a high or low resistance when you touch your probes against two points on the same baseplate (not the magnets themselves). You may need to scrape some of the wax away to make sure they can make a good contact. If the display reads 'OL' on all resistance settings, then the 'baseplate' is just the bottom of a plastic pickup bobbin. If it reads anywhere between 0.0 to say 1.0 ohms on the very lowest resistance scale, then it is metal.
1) If metal, then put the probes so that each of the baseplates is touched by a probe and check the resistance again. If it reads OL on all resistance settings, then a) the two baseplates are isolated and really need to be joined together and connected to ground. If the resistance reading is low, then b) they are connected and we only need to connect the ground wire to one baseplate.
For a) the next step would be to first clear the wax away from the metal and with a needle file then scrape away any paint on the baseplate at two positions, one on each baseplate. From your photo, the best spots would be (on each baseplate) roughly halfway between the locations where the wires emerge from each baseplate. You'll need a fairly powerful soldering iron to heat up the baseplate sufficiently to solder the ground braid to each of the two baseplates. Tin (put some solder on) the braid wire first, and it's best if you can find some way of clamping the braid wire to the baseplate (heat resistant of course).
For b) it's basically the same as a), except you only need to connect the braid to one baseplate at one point - probably where your red arrow is pointing in your photo. You can trim back any excessive braid past the solder point.
2) If plastic, then there are three choices.
a) You can ground the two magnets by soldering the braid to each of them in a manner similar to 1a). however, the magnets are rather chunky, and you'll probably have problems heating them up enough to solder the wires on without causing damage by melting plastic or melting the glue holding them on.
b) You can leave the magnets ungrounded. It is good practice to ground them, but many pickup types, especially Fender types with magnetic pole pieces, don't have grounded magnets, but as we know, this can make them more prone to picking up noise - especially single coils - which is what you will have if you turn one of your volume knobs right down. If you go this route, then I'd trim back the braid ground wire close to the grey sheath.
c) as b), but in addition, foil shield the pickup cavity with copper tape, ensure that the foil wraps over the top of the body a little in one place, ensure that there's a wire soldered between the copper tape and the guitar's ground point, foil shield the underside of the pickguard, and you should then end up with a much quieter bass than 2a) alone.
Of course if you can do 1a) or 1b) then you could also carry out the actions of 2c), for an even quieter bass.
If you don't feel up to trying any of this, then there is always the option of emailing/messaging Adam, showing him the photo and asking him to send you a replacement pickup, as it is faulty and shouldn't be like that.