I can confirm, it's a one-time service per account.
When I transferred my old PS4 commander to PC, I wanted his name to be similar to my primary PC commander's name. I was able to rename the PS4 commander during the transfer, so I still have one name change on that account. I had my primary commander's name changed via support, so he's stuck with it now. My third commander was created later, and his name was stuck in all caps (UI bug). I was able to have the capitalization corrected without using his one name change.
Bear in mind that a name change will not change the name on any exploration discoveries you've already made. I believe it will also reset that commander's cartographic data, so systems you've already visited will have to be FSS's again. All old journal files will still have the old last name, so there will be issues with third-party applications that access journal files (they will show two different commanders).
A name change can be quite a headache.