What is it called?
This device is known in French as an “Écarteur de tétine de trayeuse”—best translated as a milking machine teat cup liner expander (also described as a liner flaring/rolling tool).

Where it was used
It belonged to dairy farming operations and was used during milking on the cow’s udders (the teats/trayons) to help fit the milking equipment correctly and safely.

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Why it existed (the problem it solved)
In the early period of mechanical milking (“pots à traire” era), the rubber liner (manchon) that wrapped around the teat was not pre-molded. Because the liner was inserted into a stainless-steel tube, farmers needed to roll/flare the liner’s edge so it would sit properly on the tube and avoid hurting the udder.

Main purpose and function

Primary function: Expand/roll (“rebouler”) the end of the rubber liner on the metal tube.
Key benefit: Reduce the risk of pinching or injuring the cow’s udder while ensuring a secure, consistent fit.
Practical advantage: It was faster and easier than using fingers, especially when done repeatedly.
How it was typically used (simple steps)

Insert the rubber liner into the stainless tube of the teat cup assembly.
Place the tool at the liner’s end.
Operate the mechanism to flare/roll the liner edge into position.
Confirm the edge is smooth so it won’t rub or cut the udder.
When it was common

It was used before liners were factory pre-molded, meaning early-to-mid 20th century in many dairies that relied on older milking systems.
The text specifically notes that after World War II, liners became pre-molded, reducing the need for this tool.
Why it disappeared

Farmers had to do the rolling step often, because liners were removed after each milking to be cleaned in hot water.
After WWII, manufacturers introduced pre-molded liners, so the rolling/expanding task was no longer necessary.
As a result, these tools became obsolete and were often discarded or scrapped.
Quick identification features

A wooden handle for grip and leverage
A metal expanding/lever mechanism at the top
A design meant to open/press/shape the liner edge consistently without damaging it