I have no idea what this part is called. I want to figure out if I can remove the wire out of it and slide in the other broken end before soldering it the board. Is it reusable, or do I need to buy another one? (e.g. because it's heatshrinked or something)
You can reuse that strain relief.
Strip the insulation off the cable above the strain relief and then pull out the copper wire strands.
Then use a small drill bit and drill out each of the two halves of the cable.
Work your way up in drill size.
Use pliers to turn the drill bit.
Don't use a power drill unless you are very skilled with it.
The strain relief and the cable are usually made from different plastic, so it is easy to scrape out the cable insulation.
Once the strain relief is cleaned out, then pull new cable through. (or the undamaged old cable)
I have done this procedure few times myself.
Found it, it's a "strain relief grip" and this one can't be replaced since it seems bonded to the wire.