It sounds like you're talking about prepay plans. You "cancel any time" by simply not renewing it.
The details depend on the plan but the prepay phone plans I've used have been based around the date I signup and pay. The actual end of the calendar month hasn't meant anything. The end of the billing period is usually a month from last payment. Well, it might be 30 days. With months you can get into odd questions like "when is one month from January 31st?". Again, you need to read the fine print on the plan or ask whoever sells it.
It also sounds like this is intended for a router. Your options for prepay plans that you can legitimately put in a router are very limited, especially if you want a significant amount of data.
Here's my answer to a similar questions on another thread.
As for needing a new SIM, again it depends on the plan but it appears that the AT&T plan I mention in that link lasted a year during which time I could have restarted without a new SIM.
All that said, I think any plan these days can be canceled at any time. The days of phone contracts are gone but canceling a postpaid is a bigger deal and more of a hassle and to reactivate you need to go through the process again.
The FMCA has quite a good AT&T plan that you can pause but I think you can only pause for a limited time and only a couple of pauses per year. Unfortunately you can't even read about it now without being a member because the FMCA's website is kind of messed up. If you click on Tech Connect here https://www.fmca.com/fmcarvclubbenefits
it goes to a login page.