I came here to say something like this. I looked it up, and they actually did used to have a feature to do that, but seem to no longer allow it. I assume it’s for transparency, so you can know it’s the same person you’re working with each time, and to prevent various forms of sockpuppeting, etc.
So why not just have a display name that’s separate from the account name, and let the display name change (even once! or even by special application in the Help Centre on a case by case basis! or both!) but keep the account name permanent. Lots of social & selling sites do something like that.
If you’re wondering if the person is The Real Deal just check the account name on their main profile, instead of the display name that appears elsewhere on the site.
EDIT: To add, the reason I’m saying this is basically the above. When I signed up, I underestimated how “big” the site is. I thought it couldn’t be a big enough deal to bother integrating with my normal art socials and outlets, so I didn’t bother branding it (and the site seemed to require real names and faces. Too late did I find out it’s merely encouraged) the same way I do for eg. my FB page linked in my profile.
But now I’ve changed my mind about this place, and really wish I had my art handle to go by here. But if I close the account to switch, I lose all the business and info I’d gotten that made me want to stay. If that happened, I’d have to consider if I wanted to all. At the end of the day, I just can’t image it’s good for business for Fiverr to be encouraging people to close their accounts for any reason!
As a side note, I also overestimated how comfortable I’d be with my name and pic out there for real. I don’t know what I was thinking! I don’t even do that on FB! But I thought it was required. The pic I’ve already changed, but seeing my name up there kinda weird me out still.