No (frequently styled №) is the abbreviation for the Latin word numero (in number). Why is it also used to abbreviate the English word number? Because Latinate abbreviations were second nature to educated people who wrote in Latin. When they wrote in English, they carried these abbreviations over to the “new” language.

Other examples of this practice:

• lb, abbreviation of Latin libra (balance) – repurposed as the abbreviation for English: pound

• &, a stylized way to write Latin et (and) – repurposed as the abbreviation for English: and

Why does English use "No." as an abbreviation for "Number"?

It's a preserved scribal abbreviation like the ampersand & (formed by eliding the letters of et to mean and).

The OED has it in use from the 8th century, based on the ablative numerō used for an implied preposition in: X in or according to number.

It also gets used by the French based on numéro, which produced Wiktionary's erroneous etymology. Americans do use the # sign, although it's now often understood as a hashtag; the British upper class, on the other hand, has looked on Latin and French as providing gravitas and cachet since the Conquest.