It may well be for liquid condiment - largish examples were made - but equally it might be a decanter jug for claret or wine.
I can't see anything that would make me think this 'must be English' - those cut notches were a decorative feature from both the States and Europe during the second half of the C19. Unsure if you are saying the body only is 7", or whether that figure includes he stopper.
Occurs to me it might be a flagon (from the French 'flacon' meaning a small bottle) - smallish handled decanters for dispensing wine at the table, and the other features seems to agree - even the size.
Unusual for a quality piece not to carry matching Nos. - although that might be explained if this were for oil. The main cutting seems to be showing small relief diamonds, quite a common style of cutting.
As to the script, regret no idea, and although I'm not well up on these things have to say that I've not seen this feature on stubs before - although must admit those letters don't look especially Continental to my eyes.
Sorry this is of no real help.