How many punctuation marks are there?

There is no standard list of punctuation marks. They are designed by and for printers, not writers, and every publisher has their own set of punctuation marks, and a style manual for their use. It's been this way since printing was invented, and has never stabilized, even in (or perhaps especially in) English. So don't bother counting; somebody else could count different.

As I put it in this encyclopedia article,

Punctuation refers to various systems of dots (Lat punctus) and other marks that accompany letters and other glyphs as part of a writing system. Although there is a certain overlap among the various kinds of markings, punctuation proper should be distinguished from diacritic marks, which are intended as modifications of individual letters (e.g, ö ő å í è ç š ơ ū ñ ṇ) and are often simply considered to be part of the letter they appear on, like the dot on the Latin lower-case i. Punctuation marks are distinguished as well as from logographs, which are simply one-glyph representations of lexical items (e.g, @ # $ % &).

Several other common glyphs, like the slash indicating alternation (and/or), or asterisk ‘ * ’ indicating some special qualification like a footnote or ungrammaticality, are intermediate between these categories, but are not considered true punctuation. This article will consider only punctuation marks that appear separately from other written glyphs and have no lexicalreference.

The modern suite of punctuation marks includes the period ‘.’, comma ‘ , ’, colon ‘ : ’, semicolon ‘ ; ’, left and right parentheses ‘ ( ) ’ (and other brackets, square ‘ [ ] ’, angle ‘ < > ’, and curly ‘ { } ’), interrogation and exclamation marks ‘ ? ! ’ (bracketed with their inverses ‘ ¿ ¡ ’ in Spanish), dashes of several lengths ‘ - – — ’, single and double quotation marks ‘ “ « » ” ’, and the apostrophe, or raised comma “ ’ ” (not to be confused with the prime mark ‘ ′ ’, which is a diacritic).