There is a lot of bias against guys being bi, from "the straight" crowd as well as "the gay" crowd. Bi Males are viewed as "closet gay" or a guy can't make up his mind. Its a double standard as bi women are usually more accepted. Romantic involvement is something that is an individual desire. For me there is no romantic desire ... Just a sexual one...the romantic feelings I only feel for women. And there in lies the the key for my fulfillment...I have to have a partner who is also bi so that she completely understands me and I her...I have been lucky to have two women in my life that fulfilled my desires and needs and I theirs...when you find something like that ... It is a whole new ballgame. Society I do not think will ever be fully accepting as long as we have Religion trying to run everyones lives. Ironically their treasure trove of sexual deviance can be incredibly sick and twisted "roll the inquisition tape" and let us not forget "populating and repopulating" the earth TWICE using incest...and they want to judge US ?
It is a subject that unfortunately is still taboo and looked at with disdain. Someone a long time ago made a statement " close your eyes ... Now ... Does it really matter ? It is all about whether or not one can accept the fact that when it comes to sex...IT IS A PRIMAL INSTINCT...sex is needed to continue the human race...or we perish...it developed through evolutionary process to also be very stimulating, fun, erotic, and romantic. Once the survival of species side of it is satisfied we still have the fun, erotic, romantic and stimulating...so we enjoy as we may and take pleasure in each other in whatever satisfies us as individuals as long as everyone is on the same page...there are more than two pages that can be looked at and each is just as valid as the next.
We don't live in a perfect world but we have the right to make our own individual microcosm as we see fit without judgement from people who are narrow minded robots ...

I agree with Artoa's comment that this issue definitely tends to get aimed at bi men far more than bi women. I think the sex-prejudiced assumption is that men are generally more permiscuous than women, and therefore, by extension, any man attracted to more than one gender has that much more likelihood of cheating, while women supposedly don't. Add to that the assumption that bi men CAN'T be monogamous. It adds up to dismissive, ugly judgment. Slut shaming, indeed.
I agree also that we are all entitled to be attracted to whom we are attracted to. That Can't be dictated by others. What is aggravating about the woman you described is not merely her stating her preference, but also her disapproving tone and implication that no one....or at least no woman....in her right mind should go anywhere near a bi man. That is obviously insulting.

in my experience, biphobia is heavily based on sexism. some women and men will object to a bisexual male, but are okay with bisexual women. for some men, this may play into fantasy sex, the idea they can keep two or more women happy at the same time. i have met men that simply will not acknowledge a woman's desire and identity for same sex relationships, they are just women who haven't met the right man. most "lesbian porn" is for the cis male gaze. i think the biphobic women are threatened by bisexual men, as most i've come across are highly critical of themselves and other women for other non sexual reasons as well, so the idea that they have to compete with women and men for their affection is a brutal contest. but like the others said, bigotry is never easy to unlearn and it's just best to seek out the company of those that aren't.

The reality is, you're both right.

1. To absolutely rule out the possibility of ever being attracted to and willing to date someone who is bisexual, is a form of biphobia.

2. When it comes to who you have sex with, you can be as bigoted as you like. The only thing that matters is freely given consent, and consent out of a sense of obligation or political correctness is not freely given.

So while you're right to challenge it, ultimately you have to accept it. Bisexuality will be a deal breaker for some partners, both homosexual and heterosexual. Refusing to have sex with someone is totally different to refusing to bake them a cake or make them a pizza.