Aeons (well perhaps not quite) an ancient Greek (not Ptolemy although he wrote it down on his map) rocked up on these misty islands and after overcoming the initial language barrier asked “Well, who are you”. “We’re the Prydain” replied his hosts.
And thus British Islands, overlooking the fact that Ireland was inhabited by a different branch of the Celts. Of course Ptolemy later used the names Hibernia and Albion, and an awful lot of wasted bits would be saved if those had stuck.
But for better or worse, it didn’t stick. But also it wasn’t the English who invented the term; it was widely used amongst geographers a thousand years before the Saxons invented England (to appease the Angles otherwise we’d be called Sexland).
But to those who like to gloss over 1,500 years of history, it can easily seem like a conspiracy to claim ownership by the English. Which tends to overlook that everyone has been trying to seize power over all the islands; and it was the Scottish who succeeded in the end.
But if we were to translate “British Islands” into modern English it would be “Celtic Islands”.