Oh dear. I'm not sure I want to wade in here, because engine oils for bikes is a very complex area and I don't want to scare newbies.
The big difference between car and bike engines, which makes this area so apparently tricky, is actually in the transmission, not the engine. Whereas cars have separate lubrication systems for engine and transmission, bike engines and trannies often share their oil. Additionally, car clutches sit in fresh air (albeit hidden inside a 'bell housing') but many bikes use 'wet clutches' which sit in the very same oil as the engine. MotoMike referred to this earlier.
Obviously, oil is designed to REDUCE friction, whereas clutches rely on friction to transmit power from the engine to the gearbox. The role of the clutch is to permit this power transfer to be interrupted while gears are changed and when stationary. Consequently, it is important to use the right grade of oil in order to maintain clutch friction within design parameters.
Why is this tricky? Well some oils provide more friction reduction than others and this can affect clutch operation. Whether this change in clutch behaviour is noticeable enough to be a problem I can't say, never having experienced the issue myself. It is easy to imagine though, that high quality synthetic oil, being a 'low friction' lubricant, could have this affect. Some car engine oils also contain additives which can actually harm wet clutch friction material.
For myself, I stick pretty rigidly to manufacturer's recommendations because (1) I don't have the knowledge required to second-guess Honda in light of the above and (2) because I change my oil frequently, thus negating many of the long-life advantages of expensive oils, I would be wasting money. There is also the matter of warranty to consider, where applicable. It might be that using an oil outside of the manufacturer's recommendations could invalidate a claim if engine lubrication was implicated in an engine failure.
I also take the view that, with low-tune, unsophisticated engines such as the CBR's, fancy oils are not necessary.
If you can confirm that a 'better' oil does no harm to the clutch and you don't mind spending the extra money, then go for it. It's a simple way to 'modify' your bike and maybe release another half horsepower!
Oh and another thing (just to complicate matters further), those of you living in extreme climates (very hot or very cold) will have to choose oil with flow characteristics (viscosity) to accommodate those conditions. All oils have 'safe operating temperature ranges' and oils that flow well at very low temps may overheat and fail at very high temps (though some expensive oils have wider operating ranges). I haven't got the manual with me but all car manuals cover this, providing alternative oil grades for such conditions. If the CBR manual does not, your local Honda dealer should be able to advise.