This is an 1869 portrait of sewing machine manufacturer Isaac Singer. You see this type of robe in lots of Victorian portraits; it’s similar to the 18th century banyan.
Is there a name for it more specific than dressing gown? It’s obviously meant to be informal AND formal (or at least expensive) at the same time. (Singer was a ladies man, to put it mildly- he had more than 20 children by at least five women- which is probably why he wanted to be remembered in pre-Hefner fancy bedwear.)
The link didn’t work for me. Although I found a link through Google Images, I hesitate to post it for fear of violating copyright, I will say I think that could be a smoking jacket.
I’m pretty sure “dressing gown” is simply BrE for “bathrobe”. In The Hitchhiker’s Guide To The Galaxy, mention is made of Arthur Dent traveling around the universe in his dressing gown, which he happened to still be wearing the morning he began his interstellar adventure, mere seconds before the Vogons demolished the Earth to make way for a hyperspace bypass.
The French name robe de chambre for this garment was often used in English descriptions.
Lise Schreier’s 2010 article “Portrait de l’artiste en robe de chambre, 1830-1870” (Romance Studies, Volume 28, Number 4, November 2010 , pp. 279-295) discusses why the hell 19th-century society decided it was a good idea for portraits to show guys in their bathrobes:
I too would tend to reserve “smoking jacket” for the shorter garments, though I know that convention wasn’t universally adhered to.
It’s a dressing gown. The reason that “smoking jacket” exists as a phrase is to describe the short lounging jacket that evolved from the dressing gown.
Curiously, I learned in French class that the French call a tuxedo “un smoking.”