The pioneer psychoanalyst Carl Jung believed that we each have within us many traits of the opposite sex; and that these are represented by one or more dream figures of that sex. He named these figures the animus and anima.

It's quite possible your animus is trying to get you to know him, seeking out an appearance you can relate to and (hopefully) an approach you'll be more responsive to.

You're right. It's most unlikely that this has anything to do with your boyfriend; and it's unlikely (but not impossible) that he wants to compete with him.

That's very interesting!

The reason I especially find it interesting is that my male side is much stronger than my female, and my femininity is tiny compared to my masculinity. I don't particularly enjoy being female, and I have a bit of gender identity issue. Throughout my entire life I've been more boy than girl, and I've even given my male side a name and features as a character with which I use in various works of art.

Thinking of me running from my male side is sort of strange, but at the same time, it's not, because I try to avoid admitting that I'd rather be male (at least to people that know me) because of the discomfort it causes in people. Maybe it's just that somehow my mind is trying to find ways to make my male side unpleasant so that I'm forced to part with him to make it less painful?

Or maybe it could just be a metaphor for my desiring of my male side to take over or dominate my female side?

I'm even more curious about this, now. Especially because I tend to tell the men in my dreams "I can't because I have a boyfriend" and it's never really, truly explained what I "can't do," but it's obvious I don't want it to happen, whatever it is, because it will make my boyfriend unhappy. At first I assumed it was "I can't cheat on my boyfriend" but I'm starting to think it's because I feel I'm betraying him in some other manner. I never, ever actually come out and say "I don't want to CHEAT ON my boyfriend." I just say "I have a boyfriend, I can't do that," or "I love him."

Sometimes the man in my dreams can be friendly, pleasant, and outgoing with me, and I enjoy his presence until I tell him "I can't," which is when he gets upset. Maybe when I'm telling him "I can't," I mean to tell him that I can't let him take control of me.

In all other dreams I have aside from these, I usually am male, myself. In most dreams, I'm a guy (usually my male persona), and I'm happy about it.

Perhaps your male dream ego wants to talk about things with your female dream ego. Or, if you prefer, your female persona.

He does seem extremely eager to communicate with the female me in my dreams. He appears often in various forms, but lately he's been appearing in forms that are more familiar to me, and I've strangely reacted more comfortably to him because of it.

My experience has been that dream characters will do this. They really want to communicate, so as time goes by and they learn your comfort zone, they shape-shift to meet it.
A thought: keep an ear on your inner dialogue and see if you don't find a discussion about this going on now and then.

I actually had a dream about him again last night, and posted an entry on that a few moments ago. This time, he didn't even talk to me, he just used body language, which made it really hard to tell what he truly wanted.