I’m having the same issue, except that AdSense emailed me with the “You have another AdSense account.”
Except that I don’t. I have no connection to the email address it gave me. I emailed support and received no response.
So I marked that I had closed the old account, just to see what would happen. Five hours later I got the “another AdSense account” message.
For a completely different email that I also have no earthly connection with.
So, in a “you gotta be kidding me” reaction, I marked again that I “closed” the other account. A day later, I’ve not gotten an email telling me that I have a third, mysterious AdSense account (that I have no connection to).
I am at a loss to understand why AdSense thought twice that I had two other non-related accounts.
At any rate, Site Kit spins trying to load AdSense and briefly gives me the no Ad client found message. So, I went to AdSense and marked that the code was on my WP site in defiance of the endless install.
And AdSense said it found the code on my site. So, I guess I’m waiting a few days up to the two weeks that AdSense said it’ll take for the next step.
It’s been a fascinatingly wonky process so far.