Yes it's a feeding cup, used to provide liquid (milk, soup, posset) to infants, invalids, the elderly or infirm. Anyone who needed help with being fed. The date of registration is as you suggested, 3 April 1875, and I've searched the online registers at the National Archives and the search has returned 3 registrations for that date (I'm unsure if the whole collection is online though.)
The three are as follows:
Registered design number: 290262.
Proprietor: Edward F Bodley and Son. Address: Burslem, Staffordshire.
Subject: Tea ware basket embossed.
Registered design number: 290261.
Proprietor: Edward F Bodley and Son. Address: Burslem, Staffordshire.
Subject: Tea ware "Stork" handle.
Registered design number: 290259.
Proprietor: John Dimmock and Company. Address: Albion Works, Hanley, Staffordshire.
Subject: [Not given in BT 44/8].
The description of the first one (basket embossed) could be the one you have, but I'm not sure. It's a place to start looking though.
