I feel like this question may fare better on r/etymology

Yes, both come from the French 'Salon', and is attested to in English at least by the late 17th century to mean:

(a) A large and lofty apartment serving as one of the principal reception rooms in a palace or other great house. (b) A room, more or less elegantly furnished, used for the reception of guests; a drawing-room.

Interestingly, it was spelled both as 'Salon', 'Saloon', or 'Salloon', variously, and differences in word by spelling came later as it developed more shades of meaning, but in general descend from this very broad meaning that it is a room where people would gather.

For 'Saloon', as a drinking establishment, that arose in the mid-19th century, although interesting, while coming into use in both the US and UK fairly contemporaneously, the two have slightly different shades of meaning:

In the U.S., a place where intoxicating liquors are sold and consumed; a drinking bar. Also, in British use, a refreshment bar in a theatre; a separate bar in a public house (as opposed to public bar)

For the spelling with one 'o', by the beginning of the 1800s, it has very 'French' connotation, and is well associated with ideas of 'fashionable'. The OED gives the first use of 'Salon' in the context of 'beauty specialists' as 1913, highlighted from Vogue, and noted as a particularly American usage. So certainly it was established by the early 20th century.

All citations from the OED Online.