After a few more emails with Apple Developer Support, I finally received confirmation that my first payment/enrollment had been cancelled and that the payment would be refunded. After that, Apple enabled the Enroll button for me in the Apple Developer app. I completed the enrollment again through the app and made the new payment. The next day, my Apple Developer account was fully activated and I was able to access App Store Connect and the other developer services. Also, the refund for my first payment arrived in my bank account within one day. So the issue is now completely resolved. It took quite a long time, but in the end Apple refunded the first payment and I successfully completed the enrollment through the Apple Developer app.

Update: Unfortunately, the issue is still not resolved. Apple asked me to upload my ID and address documents again. After waiting a few days, they asked for another verification: a selfie while holding my ID. I uploaded the requested photos and waited another 3 days. Today, Developer Support told me that they made an update and I should be able to continue my enrollment. However, the Enroll button on my iPhone is still disabled. I can continue enrollment on the website, but this does not make sense because I already completed the website enrollment before, paid the Apple Developer Program fee, and received the invoice. My membership was never activated. I still have no clear information about what happened to my original payment or why I am being asked to continue the enrollment process again. I contacted the Senior Advisor handling my case again and asked for a clear explanation before doing anything else. This process has been extremely frustrating. I have completed every verification Apple requested, but after all these steps, I still do not know what happened to my paid enrollment.

Finally reached to a senior apple advisor via call. They said they will investigate and give update to me asap.

I tried to connect apple support via phone, waited approximately 3 hours of my day but still could not reach to anybody. I can not conctact to apple support by call/mail.

Still no one contacted to me for this issue. I have already paid to the Apple and still waiting for the developer access.