I don't know whether a similar trademark is a real problem

Are you selling in the same category (ceramic mugs/pottery)? If yes, they probably have a decent case. 'Clay' and 'Cozy' reversed is going to be hard to argue as distinct.

Not a lawyer, but I've been through something similar. First thing: are they actually enforcing the mark? A message could be from a lawyer or could just be the owner blowing hot air. Check if they have a registration number and look it up on the USPTO site (assuming US).

cozy clay pottery is definitely going to win that one. sorry.

Trademark atotrney here (sorry, I know we're not supposed to be on here). The real question is likelihood of confusion. Same goods, similar sounding names. You don't need identical names for infringement. Eight months of use without registration means you have common law rights, but they're limitd to your geographic area and you'd have to prove it. Their registration gives them a big advantage. I'd take the message seriously, but that doesn't mean you have to roll over immediately. Get a free consultation from a local IP clinic or a law school program.

Also, what kind of message was it? A cease and desist letter? An email? If it's a formal C&D, that's a step up from a friendly note. And did they give a deadline?

Change your name to 'Cozy Clay & Co.' and watch the chaos unfold.

If you have any proof of when you started using the name and any sales records, keep those. But realistically, even if you win, legal fees will eat you alive. Might be cheaper to rebrand now.

One more thing: had you heard of 'Cozy Clay Pottery' before this message? If not, that's not really a defense but it might help with the 'innocent infringement' angle. But again, they have the registration.

Also worth checking: is their trademark actually for mugs, or just for pottery in general? If they only registered for, say, planters or something, you might have a narrow overlap. But 'pottery' broad enough? Probably covers mugs.