There doesn't seem to be any evidence that Shaw was inspired by Shakespeare specifically in writing this line.
Shaw on Shakespeare
Shaw's opinion of Shakespeare, in general, is somewhat complicated. He's well known for being openly critical of the immense prestige given to Shakespeare in the field of English literature, and has written as negatively about him as the following:
There are moments when one asks despairingly why our stage should ever have been cursed with this “immortal” pilferer of other men’s stories and ideas, with his monstrous rhetorical fustian, his unbearable platitudes, his pretentious reduction of the subtlest problems of life to commonplaces against which a Polytechnic debating club would revolt, his incredible unsuggestiveness, his sententious combination of ready reflection with complete intellectual sterility, and his consequent incapacity for getting out of the depth of even the most ignorant audience, except when he solemnly says something so transcendently platitudinous that his more humble-minded hearers cannot bring themselves to believe that so great a man really meant to talk like their grandmothers. With the single exception of Homer, there is no eminent writer, not even Sir Walter Scott, whom I can despise so entirely as I despise Shakespeare when I measure my mind against his. The intensity of my impatience with him occasionally reaches such a pitch, that it would positively be a relief to me to dig him up and throw stones at him, knowing as I do how incapable he and his worshippers are of understanding any less obvious form of indignity.
Shaw coined the term "Bardolatry" to describe those afflicted with an excessive hero-worship of Shakespeare and his writing. Some of his works attempt to improve on Shakespeare's, such as Caesar and Cleopatra and Cymbeline Refinished. He even wrote a comical puppet play Shakes versus Shav in which the characters of Shakespeare and Shaw meet each other and engage in a fistfight followed by a debate in which Shaw tries to diminish the value of Shakespeare's contribution to English literature.
On the other hand, Shaw also campaigned for the creation of a Shakespeare National Theatre, along with the London Shakespeare League, and even wrote a short play The Dark Lady of the Sonnets, featuring Shakespeare as a character, in support of this campaign. It's noted by Trey Graham writing for the Folger Shakespeare Library that Shaw's negative views on Shakespeare may have been exaggerated for provocative effect:
Shaw was also famously a bombastic old curmudgeon (even in his youth), a genius gadfly given to making outrageous claims to get an audience’s attention, then using his peerless wit and erudition to make a watertight case for the more measured position he’d actually come to argue. (Or the opposite position: Shaw is forever giving equal time to the opponents of his own ideas.)
Indeed he would over the course of his long life shower much praise on Shakespear, to use the spelling Shaw preferred, and would even lend his name and pen to the earliest efforts to establish a national theater in England.
So Shaw wasn't afraid to directly cannibalise Shakespeare plays, but what about taking inspiration from Shakespeare in terms of language style? Well, the master's thesis Eman Mahmud Ayesh El-Abweni, "Syntactic Changes in English between the Seventeenth Century and the Twentieth Century as Represented in Two Literary Works: William Shakespeare's Play The Merchant of Venice and George Bernard Shaw's Play Arms and the Man" (Middle East University, Jordan, 2018 shows in detail that there was a noticeable difference between the writing of Shakespeare and Shaw in terms of syntax and English style. That's natural, but how about specific passages written in a more flowery style? Shaw wasn't a fan of Shakespeare's "rhetorical fustian", but would he make his own characters employ it in specific circumstances?
Arms and the Man
The evidence you've provided in your question is pretty thin for any connection between Shaw and Shakespeare in the quoted line. You suggest that character "speak[ing] poetically" suggests the influence of Shakespeare, but he was hardly the only writer in the history of English literature to use poetic flowery language! Why would you guess at inspiration from Shakespeare specifically, rather than any other English playwright, poet, or writer? In fact, even Shaw's usual language that you describe as "quite normal" is still more flowery and poetic than what we commonly see in most literature published today.
You're right that the "affianced bride" line is written in a much more formal and poetic style than most of the dialogue in Arms and the Man, but it's not the only time in the play that Sergius speaks in such a style. For example, earlier in the play (to Raina):
I have gone through the war like a knight in a tournament with his lady looking on at him! [...] Let me be the worshipper, dear. You little know how unworthy even the best man is of a girl’s pure passion!
He speaks of having found "the higher love" with Raina, and even though in the end he falls out of love with her, the "knight in a tournament" line may suggest a possible source of inspiration for his poetic style of speaking to the girl(s) he loves: chivalric romance, which pre-dates Shakespeare both in origin and in setting.
The use of the word "affianced" is what strikes me the most as being archaic in that line. This word is used three times in Arms and the Man, but a quick Google Ngrams search (from the start of Shakespeare's writing career to the present) indicates that the time of greatest popularity of this word was actually in the 19th century, during Shaw's lifetime. It might not have seemed like such an archaic term to him - or if it did, more as in "a word like my parents/grandparents would have used" than "a word Shakespeare would have used". (On a side note, apparently "affianced" is growing in popularity again in the 2010s, to a level not seen since the 1910s. I've no idea why.)
It's quite natural for characters to adopt a more flowery and poetic style when they're speaking to the target of their romantic affection. It happens even in the real world. Why do you think so much poetry is about love? Why do so many people even today make a big special occasion to propose to their partner, the kind of thing that might look over-the-top and silly if it was done for any other occasion? It makes sense that Sergius would speak more poetically when he's talking to Raina (earlier) or Louka (later) about his love for her. Occam's razor would suggest this as the simplest and most likely explanation for the flowery language in the line you quote. Inspiration from chivalric romance or even from Shakespeare is also possible, but there's no particularly compelling evidence for it.
