i had the same thought, and here's what i found out... it's linked to the actual system. here's how i found this out.
1st test. the system knows what gamers tags were on the system... my mom has a silver account, so i recovered her account on my 360 and d/l a game under my gamerstag.... i was able to play the game on my xbox under my name or her's. i then moved my tag, her's, and the game to a mem card, took it to her house, and only my tag was able to play it, but i had to be signed into xbox to play it.
2nd test. this is when i got my 360. i have two gamer tags on my xbox, mine and my fiencee's. on my original 360, both of us can play any games (games that were bought under my profile and under her's). during some testing, i took the original HDD and put it on my new HDD. on the new 360 i could only play the games that i bought on my name only when i was signed into live. I could not play the ganmes she bought, and she couldn't play the ones i bought. one more thing, I formatted the HDD of the new 360, and it asks for the serial number of the xbox. so aparently when you format the drive, it links it to the 360 it's hooked to.
conclusion: arcade games are both linked to the serial number of the 360 that originally d/l it, AND it's linked to the gamer's tag. on my new 360, i can play the games she bought, but i have to sign in BOTH of our profiles for me to play the full games she's bought and so forth.
things i haven't tried: i haven't formatted the HDD of my original xbox. this system is the one that i do the most gaming on. it is unknown if the serial number of the xbox is also linked to your gamerstag on live at the time of download. if someone wants to do this, here's what to do... (there is no need in having 2 360's to do this) create another gamerstag, and make sure it can play the full game. if you have already d/led games this will work, if not, d/l one. then format the HDD. (if you do, you will loose hexic, so make a backup if you do). redownload the games and recover the gamertags. if both can play it again, then your s/n of your 360 is also linked to the gamestag when you d/l the game, otherwise the file on the HDD is linked to the s/n number.
it's also not known if you backup your HDD and restore it on another HDD what will happen. i'm a little to chicken to do so because my wife might get mad at me since she games also if i mess up one of mine....
NOTE: I don't know if this is for every original game, but i know that with the halo 2 downloads, if you remove the HDD from one, and put it on another 360, the maps become corrupt and you have to redownload them... this is for the 360 you are putting it on, and when you put it back on your original 360.