You'll need copies of your Explanation of Benefits (EOB), which shows what the therapist is paid, and your responsibility.
At some point soon, because there are statutes of limitations for this stuff, you need to get, preferably in writing, an explanation from the practice's billing people, outlining exactly why they believe you, with your insurance plan and any other relevant circumstances, owe them a copay. Their response would need to be more than "because I say so." It should have to do with the particular insurance type, services rendered, and what the contract between insurer and therapy practice says are the correct reimbursements.
Most therapists use third-party billing services. Most of this billing work is automated, computerized. The biller may not even know what the actual terms are. They are probably entering some codes into a computer program and sending it to the insurance company. It's possible an error is occurring at that level. The billers may be using the wrong codes, either for service type or your insurance plan. It happens, but won't be discovered until somebody is willing or forced to take a close look at the details.
Insurance has the legal power to tell your shrink's office to pound sand if they are not abiding by their insurance contract. They can also retract payments from the provider's office if they determine they or you have overpaid.
This has only happened a few times in my life. One time it was a dentist's office that was engaging in wholly shady practices, and the insurer terminated their contract for what they said appeared to be billing fraud. The other time the surgical center tried to up-bill me for unbundled anesthesia services that were already covered under the original payment. What I learned from these experiences is that you must save every document, document every phone call, and be very patient for things to move slowly towards a resolution.
I don't know if you have public or private insurance or what state you're in. But many jurisdictions say if a mistake is made about in- vs. out-of-network or other coverages, if the patient acted in good faith and/or received bad information, the patient is not obligated to pay under any circumstances.
You have a state department of insurance that is a last resort if you can't get cooperation or resolution about this. You will also need to waste time talking to your therapist about how they feel this should be handled, and what they are willing to do, and how patient they are willing to be awaiting a resolution.
But the very first thing for you to do now is to send a letter with proof of service to the therapist's office and your insurance, explaining that you don't owe this copay, and asking for their assistance in resolving the billing snafu.