I couldn't erase my phone because apparently Face ID was disabled since there were issues with TrueDepth. I managed to erase it using Find My Phone from the web.
Whoever did the work on your phone did their best to keep both your own device and account out of your own hands.
The issue now is that I couldn't access with my original Apple ID because it asks me for a verification code, but I don't receive it.
That means they changed the 2FA contacts in your account. Which again was done to keep you from accessing your own account so they can control it. Proof of that is you're asked for a phone number you don't recognize.
When you erased your phone through Find My, did it restart with the "Hello" screen? If so, then it's been properly erased back to factory condition, and any garbage they installed is gone.
Rather than using icloud.com, via a web browser on your computer, can you get back into your account at:
https://appleid.apple.com/
If so, the first tab the page goes to is Sign-In and Security. It should look like this:

Change anything that is not your info back to where it should be. That would most importantly be:
Email & Phone Numbers
Password
Account Security
Notification Email
The last three are the most important to change first so these crooks don't know (hopefully, soon enough) that you are in your account removing their edits. Depending on how close attention they are paying, you may have to do these very quickly so they don't jump into your account and change things at the same time you are.
I think I would start with Account Security. Remove any devices or phone numbers you don't recognize. Make sure to add your computer, whether that's a Mac or PC so the 6 digit authentication notices can get to more than just your phone. Change the password immediately after. Then the Notification Email and finally, the Email & Phone Number section.
If you can't even get into your account this way, then about your only hope is to visit an Apple Store, let them know what happened and see if they can wrest your account away from the crooks.
As a final resort, if all of the above fails, you'll have to abandon the account and create a new one. You will lose any purchased apps and music. Also any photos and contact info you have not backed up from iCloud to another device.
Edit
Before doing any of the above, go back to icloud.com, where you erased your phone from. You'll see these big buttons at the lower left:

Those aren't just for looks. They're buttons as a way to access this data from your browser. Take the opportunity to click each one you need to get data from and save it to your computer. Contacts and Photos of course, and whatever else you think is important.
By doing this first, you'll at least have this data backed up before possibly getting into a war over control of your account.