I find that the best way to react to the dismissal of my identity (as someone who is capable of attraction to more than one gender) is to confidently restate that I am, in fact, not monosexual, regardless of the identity of my partner at that time. I typically respond to biphobic remarks by questioning the person who made them about the reasoning behind their assumptions. It seldom takes long to discover that they are founded either in a single unpleasant experience that they have allowed to dictate their beliefs about an entire group, or hearsay about somebody else's unpleasant experience. Once the source of the assumption becomes clear, the biphobic statement loses much of its former power, and it is on the person expressing biphobia, rather than the object of their biphobia, to justify the unjustifiable.
Bi phobia and bi erasure are problems that unfortunately are widely present even within the LGBT community, and the solution I believe in is the same that I recommend whenever we face cultural limitations of any kind: dialogue. It may be hard and frustrating to always have to explain to others what your reality is and what it truly means to be bisexual, but if there is a chance that anyone will be willing to listen and broaden their horizons, it is worth the effort! For some people it is just hard to understand due to internalized prejudice, but if they're willing to make and effort to understand other realities, it's worth trying! So feel free to speak up whenever you hear or read bi phobic remarks or bi erasure. Any person whose mind we can open is worth the effort.
It's unfortunately common for gay folks within the community to refuse to date bi/pan people, for various supposed reasons. I used to have a friend who held these views, and it's been really upsetting to all of my past partners (none of whom were monosexual).
When you come out as bisexual you are automatically thrown into a box labeled "different". The only people who don't look at you any differently are really people apart from the LGBTQ community. Although some people in the LGBTQ community who aren't bi have a negative opinion on bisexuals but most of them are extremely nice and love meeting new people in the LGBTQ community!
Feeling ignored and always forced into one category when you are bisexual isn't right. But this is a problem in the gay community. I would try to find some friends that are also bisexual and see how they deal with this.
Biphobia is hard to deal with as it is oppression within oppression, and it is hard to feel welcomed and accepted. Finding a loving and supportive LGBTQ community can help with this feeling. http://bisexuality.supportgroups.com/ Websites such as this one, or finding a bisexual support group in your city (if possible) can help give this sense of community.
I try my best to calmly explain to the person that they have a misconception about bi people. I've found that many biphobic people don't really understand that what they're doing is harmful. Most LGBT+ people have experienced erasure or some form of hate, so if they grasp that what they're doing is just like what others have done to them, it makes them really understand and reflect on what they've done, and do better in the future.