Is there really a voltage?
Is it possible that running so close to the live black wire makes it look like it has a current?
Assuming you mean that the wire is showing a voltage, then yes, that is very well possible especially if the wire is not connected at either end. It is not an indication of a problem.
And is it safe to use it as is, or should I crack into the wall more to separate them?
Yes you can use that wire but there are thermal limits as to how long bundles of parallel wires or cables can run close to each other. I don't think your case of two cables has a limit.
Capacitive and Inductive coupling
Some background on why there is a voltage even if nothing is connected:
A voltage can occur in a wire if
there is a nearby AC magnetic field caused by an improper pairing of live and neutral. Live and neutral wires should always be paired in the same conduit or same cable, and the neutral should carry all the return current from the live. If not, there will be time varying magnetic fields that may induce currents and voltages in nearby metal, such as nearby wires, or radiate an electromagnetic field and cause RF interference. This is akin to the transformer effect.
there are two wires running alongside eachother, close together and for some length. These pairs form a capacitor. Ineed a wire is not a capacitor, but closely spaced insulated pieces of metal (wire, metal plates) do have capacitance. The voltage (specifically the electric charge) in one will cause a voltage (charge) in the other with an electric field between the two.
This is often observed in 3-way switch circuits due to the 14/3 (or 12/3) cabling, or between relatively long runs of tightly spaced 14/2 (or 12/2 etc...).
For capacitive coupling to take place there does not have to be a current flowing. So these phantom voltages can originate in live circuit wires without active loads (breaker on but no loads), and they typically survive in other unconnected circuits or circuits with no or very small loads (perhaps a single LED light or wall wart will be exposed to 50ish V).
Is it a problem or hazard?
Capacitive coupling can occur in normal and legally wired circuits. Phantom voltage from capacitive coupling are generally not an indication of a problem.
So in your case, it is possible that the voltage in the black cable, even if the black has no loads using any power, is inducing a voltage in the white cable, even if the white is unconnected at both ends. In fact, connecting a high power load (kettle, incadescent bulb) in the white circuit, will make the phantom voltage go away, even though the load's circuit is effectively off.
You can also buy a "Stray Voltage Adapter" for your digital multi-meter.
This adapter is basically a safe way to load the circuit and confirm whether the voltage is supplied by a source might possibly hazardous amounts of voltage (e.g. a through a resistive short or strong coupling) or whether it's supplied by a weak source (e.g. short run of parallel wires)
Stray or ghost voltages occur from capacitive coupling between energized circuits and non en- ergized, non connected adjacent wiring. Because of this coupling effect and the multimeter’s high impedance, it’s not always pos- sible to determine if the circuit under test is energized or de-energized, and this creates confusion for the person performing the test.
Ref: https://www.fluke-direct.com/pdfs/cache/www.fluke-direct.com/tlk-225/manual/tlk-225-manual.pdf
For magnetic induction to take place there must be a current flowing in a wire (so there must be something drawing power) and its magnetic field must not be canceled by the same current in reverse direction in a paired wire. This special arrangement is quite rare, because the neutral -if wired correctly- always carries the same current as the live but in opposite direction. Therefore magnetic coupling is seldom the cause of phantom voltages.
Magnetic induction in wiring is often an indication of a wiring problem. Your situation does not suggest this is the case. You can use that wire for the purpose you indicate.
Physics and Math
This is a DIY site, not an electronics site, but for those who'd like a backgrounder on capacitive coupling between wires there's this:
And
Capacitive phantom voltage
Two parallel wires, insulated from each other but running side-by-side, effectively form the plates of a capacitor. If an alternating current (AC) is applied to one of the wires while the other wire is not connected, then AC voltage will tend to be capacitively coupled to the disconnected wire. If there are no other wires nearby this coupling will be nearly complete, and the voltage measured on the disconnected wire will be nearly the same as on the "live" wire. However, if there are other wires nearby (such as a ground wire in the same cable) then in effect a capacitor voltage divider may be created, and the voltage on the disconnected wire will measure somewhere between the "live" wire and the other wires.
While the voltage produced by capactive coupling may be startlingly high the amount of capacitance between the wires is typically quite low and incapable of supplying significant amounts of current
Ref: https://en-academic.com/dic.nsf/enwiki/4796365
Here's a schematic of the capacitive voltage divider, thanks to an earlier post by Pigrew in 2013:
It's that voltage divider, formed by capacitors, which brings the voltage to the 40V to 60V range, from 120V, depending on the co-bundled number of neutral and ground wires.
The specific count of wires affects the voltage divider ratio. The length of the parallel/adjacent run determines the "strength" of the voltage, i.e. whether it merely triggers a voltage detector or reading on a digital multi-meter, or whether it is a hazard to someone working on the exposed wire.
Images: