Why doesn't he want to call this a relationship? He says he's monogamous, yet is perfectly willing to carry on with this behavior when most of the time we see one another, my husband is with us, and I make no effort to hide my other relationships.
Have you tried asking him? He's in a much better position to answer that question than anyone on the internet, or your husband for that matter.
An equally valid question is: Why are you so determined to label it as a relationship? Why not just enjoy it for what it is?
My entirely presumptuous guess: It's a protection mechanism. Perhaps his romantic goal in life is to meet a woman, marry her, and be in a committed monogamous marriage. You don't fit into that plan. Although he enjoys the time you spend together, he has to keep you at arm's length in order to avoid messing up his chance to meet his future spouse.
If he's "in a relationship" with you, he cannot honestly answer "Yes" when women ask if he's single. That's an important question in the world of mono dating. With good reason, most women won't date a man who is willing to leave his girlfriend to start a new relationship; nor will they think twice about expecting him to end any casual sexual relationships in which he's engaged. Therefore, you and him "in a relationship" is not a possibility unless he wants to give up his (presumed) goal of getting married.
Maybe boyfriend doesn't know the word "monoamorous?"
Sometimes OP uses "mono" as the abbreviation where I would write "monoamorous". "Monoamorous" to me is the capacity/desire to love only one person at a time.
Other times OP uses "mono" to abbreviate for "monogamous." "Monogamous" to me is a relationship structure where it is two people in relationship to just each other.
Really, what kind of mono he is doesn't matter. This isn't a discussion about terminology. Terminology is intended to quickly facilitate communication. If you have to explain all the terms in order to have the conversation, it's easier to skip the terms and just use the explanations.
Either way, his behaviour is classically mono. He had this romantic affair† with you. As soon as he met a girl he liked and who was available for a mono relationship, he immediately ended the affair. As soon as that relationship ended, he resumed the affair. One girl at a time: mono. Emm is right that being in a relationship with a nonmono does not make you nonmono, any more than being in a relationship with a bisexual does not make you bisexual.
"a sexual relationship or a romantic friendship or passionate attachment between two people"