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)
The spell even foils wish spells and spells or effects of similar power used to affect the target’s mind [...]
Any spells or effects that are used to affect the target's mind are foiled by mind blank, unless they are more powerful than wish.
Spells only do what they say they do, but they do everything they say, and there is no flavour text.
The mind blank spell is meant to protect you entirely from the feeblemind spell.
feeblemind is not one of the enumerated effects under mind blank, but it is a spell that affects the target's mind, and as such mind blank protects you.
The list of effects that it renders you immune from is a list of examples, not a restrictive list.
Note that powers that Frighten you without effecting your mind will continue to function. The DM may decree that some Frighten effects are not effecting your mind.