hello Cathib - welcome to the GMB.
Regret I don't off-hand have a picture of the Board of Trade Registration No. 75175, but can acquire this when next I visit TNA at Kew. People here always like to see a picture of your glass item - partly out of interest, and partly as we have a member who is compiling a very useful encyclopedia of pressed (mostly) glass designs together with their corresponding Registration Nos.
From around 1842 until early 1884 designs were Registered with the Board of Trade, and the finished product usually carried a diamond shaped lozenge which contained a combination of numbers and letter from which it's possible to decipher the maker and date of Registration. After Feb. 1884 the diamond was dropped in favour of simply showing a four/five or six digit number similar to yours, and preceded, usually by the letters Rd. or Reg.
This No. isn't a trade mark........ trade marks are such things as small images of company names or invented devices/logos used by a maker to protect their designs. Surprisingly, according to Ray Slack, only five pressed-glass manufacturers in the C19 used a trade mark on their glass - Sowerby being the best known probably, and their trade-mark is the peacocks head.
Regret I don't know what the 1646 means either. Between about 1840 and the end of the C19 there were quite a few Registered Designs for pressed glass tumblers, which is possibly how your piece has been made, although alternatively it may have been blown - can you see any mould seams on the side of the tumbler? and does the underside of the base show any grinding or polishing of the glass where this might have been needed to finish the glass after manufacture.
I'm unsure of what exactly you mean when you say royal cypher - do you mean simply VR??
This design may well have been both CLASS III and IV - glass and ceramics - it does happen on occasions. The comment that this "Design is stated as a Shape of a jug in glass or china" is, on the face of it, at odds with your comment that this is a tumbler, so hoping you will do your best to provide a photograph of your glass. Of course if we had the Kew image that would help.
Sometimes a Registered design is for a type of decoration rather that the shape, and in the C19 some manufacturers were known to produce an entire service of shapes, each carrying some element of the original Registered design.
Finally, the date of any Registration doesn't mean that was the date the glass was made............... it might be, but unlikely since most Registered designs had a shelf life which continued for some years, and some examples of Registrations are known to have been made many years after the date of Registration.