I would assume insulation, protection, and identification.

Because condoms wouldn't make sense.

Play safe!

I would say identification.
The newer coils for Stern only have a small band to identify them, not a full wrapper.

If I remember right the paper wrappers are for part numbers and help with loose coil windings shorting aginst the metal bracket.

The "paper" that they are wrapped in isn't run of the mill paper. Its more like a masking tape material and it is coated. It does provide some level of physical protection but its not really insulating anything; the coil wire itself is coated which provides the insulation.

I would not use electrical tape, especially the low quality gummy stuff

Why do coils hum?
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they don't know the words.

Are not most coils made (wrapped) with wire (not bare metal) but coated wire?
Certainly flipper coils (2 wire wrap) are such.
Might it be better to wrap and ID part#s for coils with shrinkwrap plastic?
Paper tend to fall off with time or even catch fire.

Ricochet has it correct. It's printed Masking tape. Although it offers minimal protection the main purpose is as a label for the coil type.

Fun facts, Coil wire is indeed insulated, it is very similar in nature to electric motor wire which is insulated with a varnish type of product.
It is also called magnet wire. Also very close to transformer wire. All dipped or run through an insulating coating. Further since the coil itself, whether in a pinball coil or transformer coil has no moving parts they very rarely fail. The insulation typically has a voltage withstand rating much higher than rated voltage so even a small to moderate surge will not kill one, but like all things it can happen.
In electric motors for example, most motors I sell operate on 480VAC and has a withstand over 1500V.
Sorry, I'm a motor nerd, I sell them, thus I know enough to be dangerous. That is why they have me sell them not wire them, 480V hurts.