my assumption is the chromed handle should thread onto the faucet fully seating. Is there any kind of set screw or means of removing the handle part from the internally threaded section so you can re-assemble that to the faucet then put the handle back on? Are the threads on the brass part of the faucet corroded or in some other way no longer engaging?
these threads, which look a bit corroded:
Once you get the handle fully re-threaded onto the faucet, then the plastic peice w/ the o-ring moves up and down within the internal of the faucet when the handle is turned to allow a 'metered' amount of cold water to mix with the hot. If the hande isn't threaded on, any kind of water hammer from turning the water on/off within the house could pop it off.
Caveat: I'm not a plumber.... good luck.