I opened my SharePoint site (like http://localhost:3121), I am the administrator for my SharePoint farm, so I got System Account as user name. On the same system then I logged in as different user user1 to the same SharePoint site(like http://localhost:3121), then I clicked on MySite link. Now I am getting redirected to My Site of the administrator with full permission as administrator. Why it is redirecting to administrator account's MySite instead of user1 account's MySite.

When you first accessed the MySite as the administrator, were your prompted for a password? It could be that credential manager is providing the username/password for this account when you visit the website. Which OS are you running on the client machine?

And which machine are you trying to connect as User1 with? Are you using a client machine or are you logging in at the server with User1?