Let's do a play-by-play of what is happening, using the composition and the instructions as a guide:
Mix equal amounts of colorant and the developer into the bowl and mix
them well
I'd like to draw your attention to two ingredients: hydrogen peroxide (in the developer) and ammonium hydroxide (in the colorant).
To properly color hair in a permanent fashion, you need to open the hair cuticles and strip the hair of old pigment so the new pigment can bond in. Think about trying to paint a wall that was already painted before - you need to strip off the previous layer of paint, otherwise, it will interfere in the color that you want to apply.
So, the ammonium hydroxide opens the hair cuticles so the hydrogen peroxide can move in and destroy the old pigment. Since this is designed for gray coverage and not radical color change, I'm expecting the developer to be the garden variety 20 volume developer, which is not very strong.
Keep in mind that it is mixed with the pigment, not just decolorant / bleach.
Now, apply the color wherever necessary, with gloves or the applicator
comb. Keep for 30-45 minutes. Wash off with cool water.
Now the color is being applied to the hair with gloves or with the applicator comb (not the brush or comb she uses to style her hair, or so I hope) and it will be left there for 30-45 minutes.
The thing with ammonium hydroxide and hydrogen peroxide is that they're unstable. Hydrogen peroxide is even more unstable in alkaline conditions. For crying out loud, when peroxide degrades the pigment, it is also degrading itself, making it useless for another round of "pigment-eating"
Furthermore, both compounds that bleach hair are very soluble in water, so when the colorant is washed off, they'll be fully removed from the hair, leaving effectively nothing to be picked up by the styling comb or brush.
Even if there was residue on the comb / brush to begin with, if it was enough to bleach hair, it should be bleaching YOUR MOM's hair, since she uses the comb / brush more often than you.
TL, DR: no, it can't