Solutions from Apple Support:)
Two days ago, i did a live chat with apple support, here is what they suggested:
1. Reboot your device
2.Check internet connection...
3. Check for software update..
4. Reset all settings (Do it at your own risk if iphone is jail-broken)
I followed the suggestions, and of course they didn't work. So, they suggest me to have a call. (BTW my iphone is jail-broken, so it acted weird after reseting all settings)
After two days (today), apple support called me. In these two days, i kept checking game center, the phenomenon is still there. The guy in the phone spent an hour to collect my info and he gave me the following conclusions:
1. "You are using iphone 4 (ios 7.1.2) and is no longer being supported, try using a different one." (what?)
I told him there are people using iphone 6 with ios 10 have the same issue.
2. People with up to date iphone 6 can solve the problem after reseting all settings" (is it?)
After i hanged up the phone. I opened the game center again, and it is logged in with my new account in my iPhone, I am aIPHONE 4 USER andi didn't do anything. I don't know why.. But this surely contradicts with his 1st conclusion...
SO, if you believe in apple support, and;
1. you are using an iphone4, use another iDevice to do the registration and read the next line.
2. you are using a supported iDevice, do the 4 suggestions mentioned in the beginning
Hope it helps.