Rhytons are a glass drinking vessel that extends beyond the Roman Empire and the location. There are several styles, but at its core a rhyton is shaped like a horn and has animal motifs. A horn is by no means a unique design for a drinking vessel as drinking horns go back to neolithic times, and then up through Roman times, the Norse culture, and up to the Middle Ages. A curious quirk about some is that they have an opening at the tip of the horn. This allows for drinking from the bottom (see the picture below). This could be a more time specific feature but it appears in finds from not only Ancient Rome but Paratha, Syria, and Malta as well. Their purpose is, Were they used for ceremonies, festivals, or just for drinking. With the opening at the bottom, what seems to be a Roman feature/influence, the drinker was logically prohibited from neglecting their drink. However, being that it was Rome, some speculate that a servant (likely a slave) would have to stand nearby with their finger over the hole.

Speculations from various sources suggest that this was a way to show affluence, or perhaps it was for ceremonial use, celebrations, special occasion, or everyday use. As we see it associated with the household, it could likely be for something associated with the home. It’s also highly plausible that social and financial status played a role. Giving credit to the importance of the rhyton associated with the home, there are sculptures of Lares (in the simplest term, a Lar is a household deity) holding rhytons up in the air as if in celebration or at least in a position of prominence. 

Right: Glass rhyton (drinking horn), Mid-1st century CE, Roman, (Source: The Metropolitan Museum of Art /Public domain). Left: Syria – Palestine, Rhyton, late 1st century to 1st half of 2nd century CE, Roman
Drinking horn in the shape of a snail’s head. From an unknown tomb in Rabat (Mdina), Malta.
1st century CE (Source: © Marie-Lan Nguyen / Wikimedia Commons)

In Art

Roman fresco from Herculaneum demonstrating the use of a rhyton, c. 50 BC.
A late Roman-Republican banquet scene in a fresco from Herculaneum, Italy. The woman wears a transparent silk gown while the man to the left raises a rhyton drinking vessel.
(Source: Naples National Archaeological Museum/Public domain)

In symbolism: The Lares

Left: Bronze statuette of a Lar, early 1st century CE, Roman, The Lar holds a phiale in his right hand, and raises a rhyton in the shape of a dolphin in his left hand. Center: Bronze statuette of a Lar, 1st century CE, Roman, The Lar holds in his right hand a rhyton with a protome in the form of an antelope’s head and in his left a phiale. Right: Bronze statuette of a Lar, 1st–2nd century A.D., Roman
(Source: The Metropolitan Museum of Art/Public domain)

Examples of other types of rhytons from the same time period:

Left: Rhyton terminating in the forepart of a wild cat, 1st century BCE, Parthian. Right: Rhyton with the forepart of a stag, Parthian, 1st century BCE–1st century CE. Partially gilded silver, garnet, and glass. J. Paul Getty Museum, Los Angeles, 86.AM.753. Digital image courtesy of the Getty’s Open Content Program.

This is the only source I’ve found so far that sells reproductions: Historical Glassworks