Spot on. In fact, I think that without FireFox Mozilla might as well be dead, even if it survives as a legal entity and employs people. That's it's whole reason for existence as far as I'm concerned.
To the extent Mozilla sees itself as an internet company, it seems it is is so it can subsume other parts of the internet into the browser.
If it genuinely saw itself as an internet company and not a browser company, it would not have:
- killed Thunderbird, its premier non-web non-browser application;
- built a phone where every app was a browser + ad hoc web-y extensions, even where using a web application approach did not make sense;
- the exact same dumb idea as above but for IoT;
- integrated every product idea it makes or acquires (Pocket, Send, now Reality) into its browser as the primarily / solitary entry point
- continually sidelined / demoted even web-but-non-browser technologies like RSS
"Mozilla's users" (and by this you really mean, Firefox's users) see Mozilla as a browser company because everything they do revolves around their browser.
I don't think Mozilla sees themselves as a "company". Their philosophical core is as a foundation promoting core believes. The title of their website is "Internet for people, not profit" and their recruiting pitch is "We're on a mission. Join us".
Mozilla's thesis is about unlocking the potential of the Internet and ensuring it remains a public resource. Its purpose for existing is not about tree style tabs or ad blockers. But this makes for a tough process of determining how to foster innovative products since the mission is so broad.