How can I know the display manager it's being used?

You can follow command cat /etc/X11/default-display-manager. You will get response like /usr/sbin/lightdm. It looks like:

$ cat /etc/X11/default-display-manager
/usr/sbin/gdm3

The easiest and most painless way to view your environment is with the command:

env | grep XDG

Another choice is:

pstree

which will show a list of all running processes including the display manager. pstree should be installed as part of the standard distribution but if it isn't:

sudo apt-get install pstree

You can obtain more specific information about the windows and desktop environment with the following commands:

echo $XDG_CURRENT_DESKTOP

and

wmctrl -m

wmctrl will show you the name of the windows manager. You may have to install it first. wmctrl supports a rich array of fun switches.

sudo apt-get install wmctrl

These commands (except for echo $environment) all have man pages.