your empire gets no tangible bonuses like reduced upkeep for using your leader's favorite civic. neither does the AI, it's all about diplomatic relations. so when you are playing a leader, you can ignore the favorite civic for your own leader.
If you adopt the favourite civic of one or more of the AI civs you receive the "you have chosen your civics wisely" diplomatic bonus with the appropriate civs.
only while they are using their favorite civic too. the bonus starts out at +1 and then builds up over time. there's a higher limit for some leaders than for others, since some value your wise decision more.
if you change out of it after you've built up a higher bonus, and then change back into it before ages upon ages go by, depending on how long it has been they remember and give you partial credit for your earlier wise decision. so in that case, it might be that before you had +4 wise civics bonus, you swapped out, but then when you swap back you're at +2 instead of the beginning +1, which can be handy.
if you're going for diplomatic victory, or break up a lovefest between buddies for whatever reason, that "you have wisely chosen your civics" bonus can often be a good attack target. let's pretend mansa has way too high an opinion of shaka zulu. shaka's using free market (mansa's favorite civic) but that's not shaka's favorite civic so you may be able to bribe him out of it. if you can, then mansa's attitude towards shaka instantly drops. disco!
oh another thought: in BtS, you can try using spies to change their civics rather than bribes, hmmz....
