Check Firefox shortcut settings. If it's always opening in Private mode, the shortcut may include -private. Right-click the shortcut, go to Properties, and remove -private from the Target field if present. Restart Firefox normally.

And on the Settings page, you can check here:

Privacy & Security panel > History section

Either of these will trigger automatic private browsing:

• Firefox will: Never remember history
• Firefox will: Use custom settings for history +
  [x] Always use private browsing mode

Reference: Private Browsing - Use Firefox without saving history - "Can I set Firefox to always use Private Browsing?" section