If the billing cycle is 2 to 3 months, it would mean Banks have to give credit for a longer period and it makes the entire business less profitable as well as more risky compared to the Monthly billing cycle.
For example the current monthly billing cycle with a date say of 14th, means if you swipe your card on 1st day, one would effectively get a credit for 30+14, around 44 days. If you swipe on last day, one would get a credit for 14 days. On an average 22 days of credit. If we make this 3 months, the credit period would increase on an average (90+14)/2, 52 days.
From a risk point of view, on monthly cycle if there is non-payment its flagged much earlier compared to a 3 months cycle.
On offering different dates, shop around. In the older times the cycles were different, however with individuals having several cards, and trying to optimize every purchase to maximize credit period. Quite a few banks have streamlined it to monthly cycle. Shop around and some banks should be able to offer you different dates.