meaning - What do we call this kind of recursive expression?

A metaphor is like a simile.

Is there a name for the kind of statement that suggests an infinite recursion?

It is in a way similar to a paradox such as the one with a statement written on each side of a piece of paper. Side one says "The statement on the other side of this page is true"; Side two says "The statement on the other side of this page is false." But the quotation above is not a paradox, an endless contradiction, but an endless reinforcement. What do we call that?

I'm finding it hard even to come up with adequate tags for this question.

And remember: All generalizations are false.

Answer

There's no recursion here, just self-reference.

It is not recursion (and certainly not "infinite recursion") because reading the sentence "A metaphor is like a simile" does not make you repeat/recurse on anything, nor does it invoke a smaller version of the same sentence, or anything like that. It merely so happens that "like a simile" is itself a simile. So this is just self-referential at most (it's not the sentence that refers to itself, but just the part "like a simile"), not recursive.

You may also consider the category of words that describe themselves (like "pentasyllabic"), sometimes called autological (though this is for words, rather than sentences), or (imprecisely) call your sentence self-descriptive, self-similar, or self-exemplifying. ("A metaphor is like a simile" is not an example of a metaphor — the subject of the sentence — but of a simile. But "A simile is like a metaphor" would be an example of a simile.)

Something related is sentences that describe themselves, known as autograms:

This autogram contains five a's, one b, two c's, two d's, thirty-one e's, five f's, five g's, eight h's, twelve i's, one j, one k, two l's, two m's, eighteen n's, sixteen o's, one p, one q, six r's, twenty-seven s's, twenty-one t's, three u's, seven v's, eight w's, three x's, four y's, and one z.

or

"This sentence contains five words."
"This sentence contains thirty-six letters."
"This sentence contains ten vowels."
"This sentence is written in English."
"This sentence contains precisely fifty characters."

or sentences that describe a part of themselves (from Douglas Hofstadter's Gödel, Escher Bach: An Eternal Golden Braid):

"Is composed of five words" is composed of five words.

(In your sentence it's the other way round: a part of your sentence, but not your sentence itself, describes the whole sentence.)

More from GEB:

This sentence is meaningless because it is self-referential.
This sentence no verb.
This sentence is false. (Epimenides paradox)
The sentence I am now writing is the sentence you are now reading.

The self-reference in most of these is achieved by directly using "this sentence" or equivalent; in your sentence it is achieved more subtly. (So "subtle self-reference" is another phrase that may work for your sentence.) GEB has an entire chapter on self-reference and self-replication; perhaps sentences of your type are mentioned there.

More links that may be amusing: essay, list, and "This Is the Title of This Story, Which Is Also Found Several Times in the Story Itself"