….can this mismatching in charging/discharging behavior harm the battery or super cap. or both of them?
NO, there will be no side effects.
The current will be shared between the capacitor and battery for both charge and discharge. In a solar panel usage configuration as you suggest, the current from the panel will be limited and the voltage will track the battery charge/discharge characteristics.
It is however very inefficient use of the Supercaps when you simply parallel them with a battery.
Consider that a 12V battery may have a fully charged state of 13.4V and a fully discharged state 10.5V. Putting a large supercap in parallel with the battery does not change the terminal characteristics. You still would have low voltage trips at 10.5V, and still classify as fully charged at 13.4V.
The charge stored in a capacitor is: W = 1/2 * C * V^2
For a capacitor in parallel with a 12V battery the total charge in the capacitor would be:
W = 1/2 * 88 * 13.4^2 ---> 7900 Joules
But since the lowest voltage is the fully discharged level of the battery you can only access a portion of the stored energy:
W =!/2 *88 * (13.4 - 10.5)^2 ---> 370 Joules
What a waste of an energy storage capability!
Does the supercap improve the battery system, absolutely it does. Large pulse current can be provided by the supercap so it can help where the output loads are DC-DC convertors (which tend to have high current spikes) especially where the battery is close to discharged (and has a higher impedance).