Unlike driving, with boating there are not a lot of regulations that will get you in trouble with the authorities.
Number 1, would be a launch permit. This varies by body of water, but essentially this is permission to move your craft between the public road to the public body of water across a piece of land. Normally the land is owned by the State or local government, and the launch permit helps to pay for that facility. You probably need this.
Number 2, is things to keep you from dying, like a life jacket. This varies by state. I am not sure if a life jacket is going to help or hinder if you break through the ice (maybe a different question?). The people walking on the ice are probably not wearing them, it is going to come down to personal preference and historical local knowledge about enforcement.
The numbers on the sail are registration numbers, to my knowledge they are like the boats serial number. They have little or nothing to do with licensing. Kind of like every car has a serial number, but not every car has license plates or registration.
There is a whole bunch of rules that apply locally; no wake zones, speed limits, restricted area, etc. There is also a bunch of stuff designed to keep you and others alive; right of way, passing, safety equipment, etc. But the number of people on the water without a clue is so significant that in practice and the amount of enforcement is so low, that it really comes down to people just using common courtesy and good sense.
If I were to try this ice sailing, would I need to follow boating regulations?
Probably yes; the only things that really applies is paying for your access to the water/ice (Launch Permit) and things that keep you and others alive. They are all pretty reasonable, you want to do them not because they are regulations, but because they are the right thing to do.