technically, the matching (or tie) Nos. are not etched - they are applied with a stylus of sorts and in effect scratched on by hand, and it's amazing how good a fit the stopper can be on older quality decaners. If the stopper carries the number on the end of the stub, then this suggests that it hasn't been cut down - although a replacement stopper may well have been made, and this would be unlikely to fit as well as the original. However, as you say the Nos. do match, so it's likely to be the original stopper.
The age of decanters may sometimes be assessed in several ways, although not all are reliable .............
Does the glass look white and bright........does the cutting look too precise and appear to have been acid polished (the 'feel' of the cutting will lack sharpness - if you are unsure of this feature, go to an antiques source and feel the difference between old cutting and modern work).
Late C18 - early C19 decanters fequently have a 'grey' tint to the glass. Are there stones/seeds in the glass.............what is the extent of wear?
Of course, if you do find an acid backstamp, then almost certainly not Georgian.
You don't mention whether or not this has a star cut base - a feature which can help with dating, depending on the number of points.
16-point stars are supposed to indicate c. 1830......... 24-point for c. 1830 - 40............... and 32-point for c. 1840 - 1860.
And - whilst radially cut mushroom stoppers are certainly a Georgian style, they didn't have flat tops.
Appreciate Andy's comments about the Georgian 'neck rings' - but regret this feature has been copied ad infinitum and well into the C20. To me these don't have the look of Georgian rings, and large diamonds are something else that appears on work well up to recent times.
The scale cutting is quite attractive, and not that common - hollow diamonds were a Georgian feature, but I'm really clueless about dating some of these other decorative features - perhaps Kevin will have a look for you.
If you're worried about the wobbly stopper, go and buy some valve grinding paste from Halfords - it really does work, but wash it out completly before drinking from the decanter - carborundum is very sharp and can dangerous if ingested

Sorry this really doesn't help you, but thought a few ideas on dating etc. might be of interest.