It's a snippet repeated time and time again by High Priestesses at each handfasting. The Lady before us isn't just any old Goddess. She is wearing the aspect of the Goddess Bride.
I say it, when I officiate. I learned to say it from those who initiated me. I've heard those who followed me say it at handfastings. Insofar as traditions can be static in a Pagan wedding, then this is a strong one.
When it was taught to me, it was alongside the speculation that all brides are so called because of Bride. It was borrowed from the Pagan religions by Christianity. I was equally told that the groom is so named because he's a link to the Horned God - the leader of all wild things.
However, I'm always keenly aware of the propensity of my Wiccan forebears to make stuff up or misunderstand history, in order to present Wicca as a solid continuance, rather than a revival based on wisps of knowledge.
Even the most cursory investigation doesn't allow that claim to stand up. Which doesn't mean to say that a Wiccan bride cannot take on the aspect of the Goddess Bride. In fact, it fits perfectly, if you have to pick any Goddess at all, rather than letting the divine shine within her naturally.
Brighid - as both Goddess and Saint - governs the hearth, healing, poetry/stories, caring for the livestock (sheep, cattle, bees etc.), crafting, smithing and the threshold of a home. Amongst a lot of other things. She is one of the main Mother Goddesses of Ireland, even if She is most associated with the spring.
In short, Brighid is a good Goddess to have around the house. She makes the home! Which is why She can be perfectly aligned with the traditional role of a wife. Wiccans, on the whole, are far more egalitarian about home-making, but no-one can deny that women have been generally viewed as the home-maker throughout history.
Is the Wiccan Bride actually the Goddess Bride? If she's preparing to take the lead in the daily running of a home, then yes, it works. It also explains why the groom is the recipient. She's symbolically saying that she will provide food for him.
But unfortunately, Brigid has nothing to do with why we call a bride by that name, whatever my Wiccan elders told me.