Unfortunately your license is a matter of public record. As other have noted you can rent a PO box and use it as your address with the FCC. You required to maintain a postal address where the FCC can contact you.
Most of us use our home address. A few use a PO box. I have also heard of people who use a relative or friends address. I am not a lawyer and I am not 100% certain that the last option is legal, but as long as any letters from the FCC make their way to you in a timely fashion, I believe that it is OK.
On related note, when most people apply for their amateur radio license they put their Social Security Number on the application. While your SSN is not published like your address is, many people are not comfortable with handing their SSN to a group of strangers at the testing site. You can apply ahead of time to get a FRN number you can use instead. A FRN is assigned by FCC when you get your license, but you can get them ahead of time.
Here is a link, https://www.fcc.gov/wireless/support/universal-licensing-system-uls-resources/getting-fcc-registration-number-frn
by the way I am a Volunteer Examiner, one of those strange people you give you SSN to on your application. I have never heard of any VE using your SSN for anything except applying for your license.