How can I remove the side glass pane from a patio sliding glass door?
I'm guessing, but here's my suspicion based on the pics posted.  I may delete this answer after you post an updated close up pic.

Where your red arrow points - the white and black strip - I suspect that's the stop strip.  It goes all the way around the window panel and appears to lock it in place.  I suspect there's one on the outside as well.

To remove the glass, you're probably going to need to carefully remove this strip from the outside of the door frame.  Then the window pane itself should lift up and out from the bottom.

To re-secure your new pet door will include re-installing the stop strip.