It's possible, but you can't unlink an account when it's the only one linked to your profile.
What you need to do is create an Xbox or Steam account, and link that to your profile, then you can unlink your PS account, and link the other one. Once that's done, you can unlink the Xbox/Steam account.
Edit: sorry, got distracted and didn't read everything. What I said above is how you go about changing your linked account, but you cannot transfer progress between accounts for any reason.
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They can change their linked PSN account, but they won't be able to transfer their current play data from one account to another. Unless there's something i'm missing here.
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No, you're not missing anything, I just misread what OP wanted to do.
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Okay cool. 😁👍🍻