The browser is Firefox rightfully, in the sense that it run the same engine as all of the other Firefox variants. I don't know why it was labelled just as browser, but I guess it may be for simplicity's sake, coherence with the naming of the other default apps and to avoid the confusion and redundancy of having the Firefox browser running on Firefox OS.

That makes sense, but this is counter to other devices, including pictures of the fx0 itself.

http://au-fx.kddi.com/eng/products/index.html

Bookmarks, history, private browsing, and syncing are not available, which raises the confusion.

Bookmarks are websites pinned on your home screen. Private browsing and history should be available.

I suppose that history is available on the default screen, but private browsing is not. Here is a screenshot:

https://www.dropbox.com/s/weum1of1dybvsfz/2015-11-27-11-53-38.png?dl=0

I guess I'm just more-so curious why some firefox os phones have the firefox-branded browser, with an interface similar to firefox for ios and android, while some do not. Instead having a less-than-great, option-less interface.

https://www.mozilla.org/en-US/firefox/ios/

The "browser" app has changed many times throughout different versions of Firefox OS. Your LG is running Firefox OS 2.1, where the browser app was simply called browser. In 2.2 is when the change to name the browser app name to Firefox was made.