Does the mind blank spell prevent the target from being frightened by magical means, such as from the effect of a spell, or by nonmagical means, such as from a monster's Frightful Presence? Do those kind of effects affect the mind of the target?
Mind blank does not prevent the frightened condition. It prevents the charmed condition, psychic damage, and various divination effects.
The more ambiguous part of the spell states:
The spell even foils wish spells and spells or effects of similar power used to affect the target's mind or to gain information about the target.
This is speaking on the power of the previous protections: immunity to psychic damage, various divination effects, and immunity to the charmed condition. If the spell were intended to work against effects that affect the mind that are not the enumerated protections, those effects would have also been enumerated. Spells only do what they say they do.
Some lower level alternatives:
Calm emotions, dispel evil and good (against certain creature types), hallow (with the courage option), heroes' feast, heroism, and protection from evil and good (against certain creature types)