Short answer: you can't because that's not what it's for.
Link verification shows you are the owner of the target by making you include the attribute "rel". If you aren't the owner of the page/site, it is highly unlikely you could do this. Hence why it's verification that you own it.
That said, we don't put a lot of prestige into the green checks. Ppl who understand their meaning might be like "Ah, okay, that's actually their site, they own it," and that's it. NBD.
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@FernLovebond OK, gotcha.
I suppose I interpreted 'owner of the links' differently. I was thinking it was a layer of protection to guard against someone impersonating you here. /shrug
@DaveMasonDotMe
In a sense it is. Your link to your personal site, the one you own and control, uses the rel link to your profile, so it functions to show that yep, you own it, it's really yours. If someone else were pretending to be you, as say /@DavidMasonDotMe and included your legit website link to bolster their image, it wouldn't show the green check on their profile, since you didn't put it in using their username.
No worries.