hello James - welcome to the GMB.

On good quality decanters/bottles from the C20, there is often a 'matching No.' - applied by diamond point or stylus - which indicates that the body and stopper started life together ............... these Nos. are mostly, but not always, in the range of 1 to 10. If your two parts started life together, then you should find an identical No. somewhere in the vicinity of the neck or lip of the bottle - although they have been known to occur on the underside of the base.
From what I can see of the mouth piece of the body of your piece - plus the presence of a handle - I'd suggest this shape is more generally known as a ewer or perhaps a stoppered carafe - not that it matters - and probably more a reflection of where in the world the piece resides. In the U.K. there is a tendency to reserve the word cruet for wet condiments.
Unfortunately, your pix lack the sort of contrast that shows the cut pattern/design - ideally a uniform dark background with less glare, will improve what we see.
Generally, people include some indication of height/size of their glass - it won't tell us it's origin or maker, but every little helps as they say.
Not knowing whether you're nine or ninety might affect our interpretation of the period that your Grandmother owned this set - do you think it more or less likely that she might have acquired the set from within the States, or did the lady travel to Europe do you know? My only rather humble suggestion is that the notches on the arises are/were definitely a States feature