The Wire is Nichrome. For a few reasons, one it is springy and flexible. Also as noted when it is a loose fit in the sleeve it travels in, there is a substantial air gap. So even though it touches the internal wall of its sleeve in places, the dielectric is largely air, which has a lower dielectric constant than the insulation.
Also keeping it a small diameter reduces the surface area of the wire, this also reduces the capacitance.
So the whole arrangement with scope probe coax is designed to keep the cable capacitance as low as possible. This helps with the task of compensating the probe for its HF response.
The wire's resistance per foot is too low to cause any issues, its Ohms/foot is insignificant in the scheme of things, it is used for its physical properties, not its electrical properties, aside from the low surface area.
But there is a somewhat awful caveat. Nichrome itself is practically impossible to solder to (Totally unlike other resistance wires such as Isotan or Constantan) this means that at each end it has to be crimped with a small ferrule arrangement. If you think you have successfuly soldered to it , when you repaired the cable, I'm afraid that is an illusion, the wire is simply sitting inside the solder and not properly electrically bonded to it. You will simply have an intermittent connection.
If you are repairing scope probe cables, it can be practical and you can shorten the cable somewhat to get rid of a damaged section near each end. But when you do it, it pays to push the thin wire into the sheath a little and make it a tad longer/floppier than the sheath. If you don't, and there is tension on it, the first tug on the cable will break the thin nichrome wire.
Obviously, you can tell from the above, I have repaired many scope probes in my time. The motivation to do it, is that the very better probes, such as the Tek 400MHz types, such as the P6137 are expensive little devils and I wouldn't dream of throwing one out just because the central conductor broke.
