Clicking the current time on YouTube's video player will modify the URL, adding a timestamp hash to it.
Alt + click: you get a precise timestamp, including fractional seconds
Ctrl + click: the final URL is copied to the clipboard
These keys can be combined, so
Alt + Ctrl + click will copy a precisely timestamped URL to your system clipboard.
Alternatively, you can use the keyboard shortcuts:
Alt + `: adds a regular timestamp to the URL
Alt + ` `: adds a precise timestamp to the URL (_note: that's two consecutive backticks, without releasing Alt!_)
Alt + q + `: adds a timestamp and copies the result to clipboard - works with both single and double backticks
Currently the hotkeys are not configurable. Firefox captures the Alt key, so if you use precise timestamps through the hotkeys, it'll be stuck on precise timestamps even on the single backtick. Just tap Alt twice and it'll fix itself.