Update:
1) If I comment, the portion of # force logout on the current user if already authenticated from verify.php, the issue has been disappeared.
The below code is commented:
if( auth_is_user_authenticated() ) {
auth_logout();

    # reload the page after logout
    print_header_redirect( "verify.php?id=$f_user_id&confirm_hash=$f_confirm_hash" );
}

2) I checked whether it creates any issue of new signup verification, if I am already logged in. However, it did not give any issues and was able to signup. After sign up, once you save the info, it logs out. At this time, the earlier logged in session also logs out. I think, this is expected and acceptable behavior

Please share your views.

$g_login_anonymous = false; is also commented from verify.php.

With same changes, verified the following:
a) New user creation by Admin and sign up from mail link - Passed
b) New user registration from login page and sign up from mail link - Passed
c) Lost password link from login page and change password from mail link - Passed (One observation is that, during this, even another user login, he will remain logged in, means both the users will be logged in. Will this cause any issues?)

We are resolving this issue as "no change required", because it was reported against an old version of MantisBT which is no longer supported.

We recommend that you upgrade to the latest stable version [1]; if after doing so the problem still exists, do not hesitate to reopen the issue.

[1] http://www.mantisbt.org/download.php