The rule is unclear so it's up to the DM and table
The rule description itself states two potentially conflicting things(emphasis mine):
This optional rule replaces a character’s proficiency bonus with a proficiency die, adding more randomness to the game and making proficiency a less reliable indicator of mastery.
Instead of adding a proficiency bonus to an ability check, an attack roll, or saving throw, the character's player rolls a die.
The first quote suggests that it is a wholesale replacement of the proficiency bonus with a die, but the second quote only talks about ability checks, attacks, and saving throws.
So which is correct?
It's unclear without an errata because each sentence seems to state something different. Whether or not the 2nd sentence is "more specific" isn't clear. Maybe they're just specifically calling out the times when you generally roll so you don't forget - but it does seem like an oversight not to discuss how to handle DC.
But that's what it is - it's an oversight over a particular rule.
That leaves it generally up to the table, but my table has used the proficiency die in setting our DCs.
DC and proficiency die in practice
Honestly, my tables never even noticed that they didn't seem to specify DC. We've been using the proficiency die at my tables for over 6 years and we love it. The non-guarantee and swinginess (and the extra die roll) are something we very much enjoy. It's never clear how well your proficiency is going to work and we like the in-game and at-table results from that.
Using it for setting DCs has been fine. While the times we roll a 1 are a real bummer, the times we roll max on a die make up for it. And the idea of "Just because you're proficient doesn't mean you do it the same each time" generally makes sense for us.