And when I login as Student I only see this as shown in image.
ok, you've created a custom menu, but did you assign it to the course?
Concerning picture 2 => logging is as an admin on the LMS does not mean you are an admin in the course. If you are an admin in the course you will know before entering it (it's written on the course miniature in the "my courses" page) and you will also see other menu items in the dropdown menu.
In summary: you are seeing the menu you are using, with the permission that have been selected.
Check and correct your custom menu (every item) and assign it to the course.
If I am login as an admin I can see an administrator in the course miniature as you can see in the picture.
You mentioned "admin will also see other menu items in the dropdown menu." but I can't see any other menu item except the menu assigned to the course even If I am login as an admin.
It seems the only menu I can see which s assigned to course even if the user is admin he doesn't have full right
Refer the picture for customer menu and its assigned module