Are both methanol and aq. Methanol protic solvents and If aq.methanol is protic the why cuz it breaks as CH3+ and OH-?

In chemistry, a protic solvent is a solvent that has a hydrogen atom bound to an oxygen (as in a hydroxyl group) or a nitrogen (as in an amine group). Therefore, methanol (CH₃OH), water (H₂O) and aqueous methanol (the mixture of CH₃OH and H₂O) are all protic solvents.

It is hardly for a methanol molecule to break into CH₃⁺ and OH⁻ ions. On the other hand, the O-H bond is easier to break. For example, when putting a piece of sodium metal into pure methanol, the O-H bonds in the methanol molecules will break to form methoxy ion and hydrogen gas.
2CH₃OH + 2Na → 2CH₃O⁻Na⁺ + H₂