Does 'moonlighting' mean 'illegal work'?

While there used to be a criminal meaning for the word, that connotation is long gone:

Moonlight is the reflection of the sun off the moon's surface — a clear sky and a full moon provide brilliant moonlight. Smugglers hate moonlight. If you moonlight, you work a second job, and it doesn’t have to be at night. If your history teacher also works as a mascot for a baseball team, he moonlights as a dancing bear. As a verb, it used to mean “commit crimes at night,” but now just means “to work a second job.” — Vocabulary.com

Moonlighting is legal (with a few exceptions that aren't that important here).

Note: that's not the same as moonshine, which is typically a noun (but is sometimes used as a verb).

The phrase you are looking for is "work under the table":

To work in exchange for payment that is not officially documented and has not been taxed. — Farlex via TFD

That is illegal (as far as I know).

It originally referred to the Australian/Irish expression meaning to steal cattle; thus moonlighting noun.

It was later used to refer to:

(UK Und.) to engage in criminal activity at night.

• 1942 [US] Berrey & Van den Bark Amer. Thes. Sl.
• 1949 [US] Monteleone Criminal Sl. (rev. edn).

and to probably illegal but not criminal jobs.

(orig. US) to work at two jobs in order to boost one’s income. The second job is usu. night work, and the other employer may not know about it; thus more usual as the noun moonlighting.

• 1970 [US] E. Tidyman Shaft 35: Cops were moonlighting as armed cab drivers.

• 1978 [US] L. Kramer Faggots 116: He moonlighted, writing twice-monthly features.

(GDoS)

Moonlighting means a second job.

That's all it means.

This is remarkably unarguably, straightforwardly, stated in the Oxford English Dictionary:

verb (past and past participle moonlighted) [no object] informal have a second job in addition to one's regular employment: many instructors moonlight as professional consultants. - The OED.

For example, a very famous TV show (actually, the funniest TV comedy ever made in the US) was named Moonlighting - a fashion model ended up working in a completely different field, for comic reasons.

(Please note that on this site, extensive discussion about the origin of, or previous meanings of, a word - when the question is "What Does This Word Mean" - lead to confusion.)

Regarding Schwarzarbeit.

In English you simply say "working off the books" or "working for cash" or you might hear "cash in hand", "under the table", "off the books"; the phrase "the black economy" is used to describe that overall economy. There's no single term for working "cash in hand".

(Moonlighting is utterly unrelated and does not suggest it in any way.)

Others have given the correct denotation of "moonlighting" currently - a second job, one that possibly your main employer doesn't know about. For tax purposes, it definitely is different than being paid under the table (by your main or secondary employer) in that it's legal work and paid and taxed legally.

However, many employers take the same sort of dim view of "moonlighting" as the taxman does of "Schwarzarbeit", and if they find out about it, it will negatively impact your main job (overlooked for promotions, not given plum work, on the list for being found not performing to expectations: "he always seems so tired"). If you moonlight in the same field as you're being paid for, they may be in the right, or at least able to restrict it in the contract. If it's totally different, the company's actions may not be legal, but it happens anyway, and you'll never be able to prove anything; and companies where this attitude holds are ones where you really want to keep your moonlighting on as much of a lowdown as you would if your moonlighting job was illegal or being paid "under the table".

As Mari-Lou A and Ruadhan2300 mention in their comments to the OP, "moonlighting" implies not just working a second job, but keeping it secret from someone (usually your main employer or the public) - and frequently having a reason to (the policeman moonlighting as a drag queen, as mentioned in the comments, or someone working for an employer known to frown on "hobby jobs").