I just can speculate...
Alexandrina was choosen to honoured the godparent of the Kents baby...the tsar Alexander I of All the Russias. The king George IV was really unfair to his brother Edward and his siter in law Victoria the day they christened their newlyborn daughter...Edward suggested a lot of names connected with the english royalty, as Charlotte, Georgiana and Elizabeth, but the king declared that she would be named Alexandrina for the godparent. And, then, he suggested -perhaps ordered...- the second name would be the same of the mother.
But Drina never liked the russians...I imagine that, while she grew up, she heard a lot of coments about the ill-fated Juliane of Saxe-Coburg, sister of her mother. So, when she refused the first name and adopted the second name, she was rejecting the russians and, by the way, honouring her mother. At this time, Drina was not so lovely and gentle with her mother...but maybe it was a little confort to the dowager duchess of Kent.