A Google search shows that this is common terminology.
Just because it's strange to you doesn't mean it is strange to the world.
Also a minus bi is often called the COMPLEX conjugate. That is to differentiate from the usual conjugate, I guess.
A comment asks if any books use this terminology. Well Google books shows several. Barron's pre calculus is a recent one. But also several algebra books from the late nineteenth and early twentieth century. Note that I'm excluding usage in terms of complex conjugated or binomial surds. But even with the very frequent usage of binomial surd, it would seem that the qualification surd differentiates from normal binomial conjugates.
I'm not sure the usage is most common. But it is certainly not uncommon. Really, what we are getting is a sounds strange to me from questioner. And Google is a helpful tool to show frequent usage. And I would caution questioners and answers to generalize from own personal experience. Sounds strange to me doesn't mean strange to the world.
For FB, well would a rose by any other name smell as sweet? We are not even kvetching about divided by zero. We are discussing terminology. And if people use the term such, how can that be wrong. Especially if common. It's just language.
Heck. My dictionary, lists one of the definitions as pairs having features in common but opposite in some particular. Further, the terms COMPLEX conjugate or binomial conjugate SURD each imply some need to further define the term conjugate...that is referring to binomials with imaginary and real or rational and sqrt irrational parts. I guess...as opposed to other sorts of conjugates.
So certainly a binomial conjugate, used as Sue and many other English speaking math teachers, for over a hundred years, is reasonable usage. It's not bastsrdizing the language. It fits fine with other uses if the word, is well defined in class, is supported in dictionaries, is common enough to be frequent, at least. Not saying it is majority though.
So anyhow if the usage is common, what is your problem with it. A rose is a rose by any other name. Oh...and a spade is a shovel. 😀
And as far as the opposite binomial surd, for rationalizing, so what? Standard method works as is and is easy to learn. Why complicate?