Learn Russian: У любых успешных людей есть деньги.

They are in the genitive plural because they are part of the phrase following the preposition У (which means 'at' or 'by'). For plural adjectives and words acting like adjectives, the genitive ending is -ых (or -их after certain consonants).

I know успешный is an adjective, but is любой (any) an adjective too?

Любой is technically a pronoun, but it behaves and declines exactly like an adjective. In the nominative plural, it is любые (any), which becomes любых in the genitive case, matching the adjective успешных perfectly.

Why doesn't the noun людей (people) also end in -ых?

Nouns have their own set of rules and endings for the genitive plural, which are completely different from adjectives. The -ых and -их endings are only for adjectives and pronouns. The word люди is an irregular noun, and its genitive plural form is людей.

Is the word есть strictly necessary here?

Yes, in this context. The У + genitive structure literally translates to 'At any successful people...' but you still need есть (there is / are) to state the fact that the money exists. Leaving it out changes the meaning and makes it sound like you are just setting a scene rather than talking about possession.