This was in the NYS Dept of Agriculture's website:

Any dog control officer or peace officer, acting pursuant
to his special duties, or police officer in the employ of or under
contract to a municipality shall seize:
(a) any dog which is not identified and which is not on the owner`s premises; and
(b) any dog which is not licensed, whether on or off the owner`s premises

So I guess you can lose your dog either by having too many or by having unlicensed ones. I have never seen a law in MY state limiting the number of dogs you are allowed. Not saying it doesn't exist but I have never heard of it before in my area. So I would rather be safe than sorry and just license them...

Here is the link to the website, lots of info on licenses there!

NYS Dept of Agriculture