No, Chrome and Firefox nowadays don’t allow extensions to be automatically installed and enabled. Any extensions running were specifically enabled by the user at some point.
I know this because I helped someone install McAfee antivirus a few days ago. Both Chrome and Firefox showed a small, non-modal popover on the next launch saying that an extension had tried to install itself. The popover contained two buttons, one for enabling the extension and one for keeping it disabled (Chrome) or uninstalling it (Firefox). If you never clicked inside the popover (as many alert-blind users might do), the extension would stay disabled.
I know this because I helped someone install McAfee antivirus a few days ago. Both Chrome and Firefox showed a small, non-modal popover on the next launch saying that an extension had tried to install itself. The popover contained two buttons, one for enabling the extension and one for keeping it disabled (Chrome) or uninstalling it (Firefox). If you never clicked inside the popover (as many alert-blind users might do), the extension would stay disabled.