I failed to log in, but my mobile device shows an alert saying that I have logged in successfully on my PC.

The log-in process in the PC version consists of two steps. You are experiencing this problem because you completed the first step, but failed to complete the second step. 

Once the first step is completed successfully, a log-in alert is sent to your mobile device. However, you may still fail to complete the second step because network administrators at banks, government offices, public institutions or certain companies may block the use of messenger programs such as KakaoTalk for security reasons. In this case, you cannot use KakaoTalk.