Thanks for that, I didn't know it! It seems that also F3 works, and in fact CTRL+G is alias of F3, and both work in Firefox as well.
The only issue is that in Firefox, it is only equivalent for the first search; once you close the bottom bar, subsequent F3/CTRL-G just do "find next occurrence" and do not display the bar anymore. Chrome always displays the search input on the other hand.
Edit: since talking shortcuts, in Firefox ' (apostrophe) is like CTRL-F but searches only hyperlinks (and you can cycle through in case of multiple matches with F3/CTRL-G) which is extremely useful for quickly navigating pages via keyboard only.
Ctrl + G certainly is hookable[0], folks just rarely know that it's an alias for 'find'. If you REALLY want the browser's search, in Chrome, you can use the mouse to open the menu and choose "Find". You could also use any keyboard shortcut that focuses the URL bar (so keyboard events are no longer sent to the page) and press Ctrl-f then.
0: In Google Sheets, for example, Ctrl-g opens the JS-driven find bar, or, if it's already open, advances the match.