I see that there is a field in the registration form that requires my SSN? Do I need really to give it to online betting site?
Every legal website in US requires this information. It's part of the requirements to be legit. They need to verify fully the information because they are responsible for its customers. This field is required and sometimes you may be asked to submit it later. The whole process is explained very well and you can receive further information from the customers support of your online bookie.
Remember that if some requires your SSN, it's not a guarantee that this company is legit. You should check it first. Some 'fake' sites not licensed in USA may try to be very smart but it's serious crime so just more responsible when you share confident information.