Can aniline be made with the method described in Dr Stone?
Believe it or not, aniline was actually isolated from coal tar way back in 1834. A chemist called Friedlieb Runge isolated aniline along with phenol from coal tar. He discovered that aniline produced a beautiful blue color when treated with chloride of lime (calcium hypochlorite). He named it cyanol/kyanol and that was the birth of the first aniline dye but it was made a dye by the exhaustive work of another chemist, William Perkin.
Perkin distilled the components of coal tar like benzene, aniline, toluene, anthracene, phenol, naphthalene and other aromatic hydrocarbons. He first worked around toluene to make allyl toluidine which he then transform to quinine by oxidizing it with a mixture potassium dichromate and sulfuric acid. He was left with a useless reddish-black powder. He then took aniline and repeated the experiment. He was left with a black gooey substance. Disappointed by the results, he took the test tube and washed it off but then came out a beautiful looking purple residue which hung off at the wiping cloth. He named the dye &quot;mauveine&quot;. Perkin was so amazed by the result that he started working with other distilled products hoping to make other dyes$\ce{^{[4]}}$ . His discovery revolutionized the dye industry. Perkin proposed the following statement:

Benzene, toluene, and other components of coal tar were colorless
because they absorbed ultraviolet light undetectable by the human eye.
But if those aromatic hydrocarbons were treated with an acid to create aniline or another amine, after some additional steps the
newly synthesized molecules very efficiently absorbed light particles
from specific wavelengths in the visible spectrum.

So, Senku (the main character of Dr. Stone) being well-versed in chemistry knows what he's talking about.
Notes and references

https://trc-leiden.nl/trc-needles/materials/dyes/aniline
https://www.scientificamerican.com/article/toms-river-excerpt-on-aniline-dye/
https://www.cavemanchemistry.com/cavebook/chtar.html
Magentas from toluene, reds from anthracene, pinks from phenol, and indigos from naphthalene.
Dr. Stone is an amazing anime. I recommend fellow readers to check it out.