Albert controls a renown charater and there is a power on that card. Bob somehow takes control that renown character. Does Bob also gain that renown power for himself? I've checked FAQ and couldn't find any obvious rule about this situation. So, what's your opinio?

From the core set rules book:

"The game is immediately over when one player
has claimed 15 or more total power between
his or her House and/or characters in play. That
player wins the game."

"His or her characters in play" always means "the characters you control." So it is control of the character that matters when counting your victory total.