This satisfies some prereqs, but it's ambiguous for others.
"Divine spellcaster" isn't clearly defined.
The game doesn't clearly define a "divine spellcaster." The closest it comes to defining it is as part of the divine spells section, which can't be taken as comprehensive at this point: there are multiple divine spellcasters that do not prepare spells like this. Prerequisites around spellcasting broadly fall into two buckets:
Faith Magic is unique (as far as I know) in terms of granting the ability to cast a spell outside of your typical tradition (without it just being an SLA, which doesn't satisfy most prereqs). The definitions of an arcane/divine spellcaster are based on the classes themselves, which implies that it's an innate characteristic of the class (i.e., a wizard is just an arcane spellcaster). Ultimately the rules do not make it clear as to whether a wizard who can cast a single divine spell is both an arcane and a divine spellcaster or if they're simply an arcane spellcaster who has a divine spell. Some elements of the game such as Divine Defiance reference "levels in a divine casting class" which lends credence to it being an inherent aspect of a class, but there's nothing concrete either way.
There are other aspects that reference being an arcane/divine spellcaster (such as Divine Strategist, an archetype; Talisman of Pure Good, an artifact; or Atheist Abjurations, a feat) which would be impacted by this ruling.
"Able to cast spells" is more clearly defined.
Divine Dignity, for example, requires the "ability to cast divine spells." Mystic Theurge requires that you are "able to cast 2nd-level divine spells" (note that "spells," in spite of being plural, does not actually requires the ability to cast multiple spells per this official rules response).
Faith Magic allows you cast a spell as a divine spell, so it unambiguously meets the criteria for Divine Dignity or Mystic Theurge (with the caveat that while it requires a higher-level slot to prepare it does not raise the level of the spell, so you would need to be a 7th level wizard). Whether it also allows you to use a Talisman of Pure Good is ambiguous.