When it seems like people are acting against their interests, people have probably failed to understand their interests. Re-reading Garam's explanation, originally the fight between princesses was about status, about wealth, about dominance. Bodies then began piling on up on one another, and soon the 10 families fought one another just to kill each other out of spite. I think there's another ulterior motive Zahard has though. When Yuri activated two weapons, the spirits mentioned they were awaiting "the chosen one", the one who will unite all 13 weapons. So Zahard's other motive is to stall for time until this "chosen one" appears.
Regarding the princesses allying with one another, it isn't surprising. Given what we know thus far, they're probably working together to oust a common enemy. Also, the infighting isn't always stable. At certain points, a princess gains the upperhand, acquires 2+ weapons, and temporarily puts an end to the squabbling. We know this because 5 princesses went on a rampage prior to Yuri's possession, and no matter how powerful a princess is, it means nothing if everyone else gangs up on her. So they had help in securing the 13 monthly weapons. Furthermore, do all the princesses care about winning the competition? Yuri and Androssi don't seem to care about coming out on top. They're interested in doing other things and will ignore the political infighting as long as they're left alone. Other princesses just appear to be interested in the status and opportunities being a Princess of Zahard brings. So they're content with what they have and have no aspiring ambitions outside of maintaining their life of privilege. Consequently, they partake actions that appear to conflict with their interest, but in reality, are actually working toward their self-interest. They're just throwing their lot in with what they perceive to be a winning party. That, and the fact that some princesses are probably more loyal to their families than they are to others, like the twins. We can speculate more on their individual motives, but these seem to be the most apparent motives from what we've witnessed thus far.
Things are beginning to change, however. For a long time, factions maintained the power-of-balance between the 10 great families, but with Baam's alliance with Yuri and Garam, this balance is beginning to crumble. In addition, Eurasia Enne Zahard will make an appearance at some point. Why? Because Baam needs Eurasia's Colorless December to unite all 13 weapons, and an extra powerhouse like Eurasia is needed to grow Baam's military power. Many people have political and financial incentives to maintain the status quo, and when Baam's group is perceived to be a threat to this balance-of-power, many of the warring factions will set aside their differences to ensure Baam's group doesn't upend their stranglehold on the Tower.