Do Alkaline batteries, $\mathrm{NiMH}$, rechargeable alkaline, Lithium $\mathrm{AA}$, all have similar MAX POWER? So in physics, one smart teacher told me, 
$$
V = IR
$$
or
$$
I = \frac{V}{R}
$$
but it is not always true, because $P = VI$ and each power supplying device (such as battery) has a "maximum power" it can provide.  So $I = V/R$ is true as long as $P = VI$ doesn't exceed the maximum power the battery can supply.
So will old style $\mathrm{AA}$ batteries, Alkaline $\mathrm{AA}$ batteries, and $\mathrm{NiMH}$, rechargeable alkaline, Lithium $\mathrm{AA}$ (Energizer) all have similar MAXIMUM POWER?  I am suspecting my Mario Kart Remote (fit in a wheel) for playing Mario Kart Wii not behave the same when using different batteries.  The best seems to be the Duracell Ultra Alkaline, and then the Duracell rechargeable pack for Wii remote ($2.4V$ only) works also flawlessly (but can it be, because it is very snug?)
Otherwise, sometimes I turn left and do a drifting and the console thinks that I am drifting right... so it does seem that different batteries have different outcome...  and I am suspecting can the Wii need more power than some batteries can provide?  and so how do they work really?
 A: For the purpose of this question, I'm assuming by MAX POWER you mean the maximum peak wattage - or in other words the intersection of voltage and current draw where power is at its peak.

So will old style AA batteries, Alkaline AA batteries, and NiMH, rechargeable alkaline, Lithium AA (Energizer) all have similar MAXIMUM POWER?

No.  They have different chemistry, and the power curve is different.  But that doesn't matter, because the root of your question doesn't depend on maximum power:

I am suspecting my Mario Kart Remote (fit in a wheel) for playing Mario Kart Wii not behave the same when using different batteries.
...sometimes I turn left and do a drifting and the console thinks that I am drifting right... so it does seem that different batteries have different outcome... and I am suspecting can the Wii need more power than some batteries can provide? and so how do they work really?

The Wii remote should consume between 50 and 200mA, or 150 to 600mW.  All these battery chemistry types are perfectly capable of meeting these requirements while they are new and /or fully charged.  As the power is consumed, the batteries will eventually be unable to meet the 600mW power requirement, and the wiimote will generally shut off to prevent players from using the remote when it is unable to operate correctly due to low power conditions.
In other words, changing the type of battery will not change the operation of the remote.
However, if you believe this to be the case, there are instructions for connecting your wii remote to a computer and logging the data stream.  You can therefore construct tests to verify your theory that different battery types will affect the performance of the wiimote.
A: You can estimate that ALkaline ones have resistance 0.1 Ohm, NiMh 0.01 and LiIon 0.005, so in short pulses you should be able to get 33W, 144W and 450W of power in short pulses, but your mario cart should not eat THAT much. 
BUT half-discharged Alkalines became 'weak' much faster(their internal resistance increases rapidly) than NiMh/LiIon, and LiIon is the very best by far.
A: All batteries have internal resistance which must be added to the R term for any circuit.  This value depends on the type of battery as well as its level of charge.  I found this page about batteries which explains the effects in detail.
The answer is NO, because P=V^2/R and R varies.
The wiki page about the Wii Remote says that a "3300µF capacitor provides a temporary source of power during quick movements of the Wii Remote".  This suggests that the device may have power requirements which exceed those of the battery; however even if the internal resistance of the battery is as high as 1 Ohm, this capacitor can be charged in 3.3ms, which is too small to account for your perceptions.  Maybe the errors are caused by low power but I don't think t = R C is telling the whole story.
A: The maximum power of a battery depends on the internal resistance (which gives the maximum current you can draw) and the chemistry which determines how long you can draw that current for before the battery is damaged.
You probably aren't interested in how much short circuit current you can draw from a battery for a millisec before it's destroyed - the maximum continuos current is more complicated.
Roughly NiCad highest, then Lithium ion, then NiMH then Alkaline.
But in terms of stored capacity (how long you can use them for) then Li-Ion, Alkaline then NiMH/Nicad
