I frequently have peeling on the inside of my cheeks where my top and bottom teeth meet, and on the cheeks near the bottom row of teeth, not on the gums. Has been occurring for years, though not painful. Also suffer from anxiety and biting my cheeks?

Cheek biting. Biting on your cheeks, clenching, and grinding your teeth will cause the tissue to peel. See your dentist for evaluation.
See DDS. See your dentist for a clinical evaluation. Possible linea alba, cheek biting? Parafunctional habits such as cheek biting can result in the situation you describe.
Peeling tissue. The peeling tissue may be caused by chronic cheek biting. Other causes may be mild allergic reaction to something in your toothpaste, mouthwash or sometimes foods. See a dentist or oral surgeon about this. They can help you find out what is going on. If you are not getting treatment for your anxiety yet, please do so. Good luck in the future.
Morsicatio Buccarum. also called chronic cheek biting. See your dentist for evaluation and habit mouth guard fabrication. Very common issue. Good luck!
Linea Alba. Anxious folks tend to grid their teeth when they sleep, even during the day especially when concentrating. It's common to get a white line that follows the shape of the line made where the teeth meet. This added to actual cheek biting does create peeling gums. Some people also have certain conditions where the gums slough off to different degrees. Usually just benign findings.
Result. The peeling tissue may be as a result of your cheek biting. Repetitive traumatic insult to soft tissue may cause annoying to serious results. Please have an Oral Surgeon examine your tissues. Please seek counseling for your anxiety.
Not sure what. the question is. You already know what is causing the problem. Have a dentist check the areas regularly... constantly traumatized tissues can convert into cancerous or precancerous tissues. Try meditation and yoga to help with anxiety.
Cheek bite. What you are describing is most likely cheek bite. The tissue on the inside of the cheeks turns white and sloughs off. You should be seeing a dentist at least once a year and have an oral cancer screening performed if you are concerned.