In some subtle ways, yes, I do think so. The English - speaking Beverly, at her foundation, is still the same Beverly as the French - speaking one (or the German - speaking one). Folks in one context might not recognize her in the others, though, due to a few factors:
First: Cultural Underpinnings. There are different codes / rules of engagement in each environment. I don't even bother trying to fit into the different milieus, but have probably subconsciously picked up some of the more important points during my various stays.
Second: Memories in the different environments. The Paris Beverly differs from the Grenoble Beverly, the Geneva Beverly, the Brussels Beverly. She's also a different animal from Herodotus Beverly (the Beverly who gets her fix of the Atlas Mountain Accent in "No Go Zones" that she manages to find during her peregrinations). She's only just briefly touched on Acadia, and would like to learn more about that, perhaps with the help of her Carthaginian Transplant friends.
Third: How the language itself structures reality. Beverly grew up with a hybrid of Polish / Russian / German / etc thinking that it was German. As no German classes were available in her High School, never got to sort things out until much later. (Thank heavens for Latin! Else, she'd not have been the B student she was in that language in college.) Learned a serious appreciation of the organisation of both languages; it's a Spartan Beauty.
As for French, you'd be surprised at how much of a reality shift it is to go from "wondering" to "asking one's self," or from "dropping something," to "making something fall."
Then there's the whole question of Time (the far past, the simple past, notions of the future), both in reality (with or without conditions) and how one's feeling about things. There's always going to be a struggle with the subjunctive and preterit in English; Latin languages make this a positive dance. German's a Whole 'Nother Animal.
During the first run of my Coursera course on the bilingual brain, a student asked whether changing languages leads to people changing personalities....