There are few clues in the photographs of this item as to its purpose. It has a body, which can be filled with a fluid substance when the top is unscrewed and a plunger with markings, indicating the amount of fluid to be dispensed. The tip of the syringe is hollow to allow the fluid to be discharged, but is not sharp and is not grooved for the fitting of an alternative tip, such as a needle. It is silver in colour, but has been well used and brown colour is showing through, so the syringe may be made of brass, but plated with another metal.
So, what was its purpose? It is not for vaccination into an arm and certainly not for icing cakes! It bears a resemblance to vetinerary instruments from the 1950s for inseminating farm animals. In this case the fluid in the body of the syringe would be semen, which would be injected into a female of the species.
However illustrated below is a very similar syringe, dated 1927, which formerly belonged to an Ear, Nose and Throat Physician at the National Jewish Hospital and this was used to remove ear wax.
The body of the syringe would be filled with warm water, or oil and the liquid injected into the ear canal to soften and flush out the unwanted wax. Care would need to be taken not to touch and damage the ear drum. The liquid would flow back out of the ear with wax. Withdrawing the plunger might also suck out loosened wax.
Excessive wax can lead to hearing loss, headache, tinnitus and vertigo. Nowadays many people use cotton buds to clean their ears, but this is not medically approved, especially if wax build up is a problem, as the buds can push the wax in, making matters worse. Various treatments for removing ear wax at home are readily available from pharmacies and syringing may be a last resort, but, hopefully, not with a syringe like this one!