Well, since there is the first level feat Ancestral Mind which moves innate castings to psychic spellcasting (and other casting classes have the same usually at a slightly higher level), I'd say it's possible, however, other traditions and innate take a bit of effort. The existence of the feat means it's not a free transfer for innate.

My assumption would be that spells from other traditions (Arcane, Divine, Primal) are cast in a slightly different way (they use a different DC and attack roll normally, which shows different skills and methods are involved in the learning of the spell). That means psychic spellcasting may not apply by default (as Occult is the only psychic tradition). GM ruling or clarification from Paizo required I'd say.

The difference with Occult spells is that these are the same spells used by the Psychic cast in the same tradition as the Psychic, just using different spell slots. This would of course only work when they are spells from an archetype being used by the Psychic class character, as that makes the Occult DC and attack roll the highest available, and the psychic spellcasting is only highest when the class is Psychic. Otherwise, the class occult spellcasting from the class (say Bard) would be higher {exception being where IQ is much higher than charisma and IQ is used for a Psychic archetype}. This is because spellcasting proficiency rises faster for class than archetypes.

Since the default for all other occult casters is Charisma (exception Witch, a prepared caster), and a Psychic can work from Int, you get a spellcaster DC based possibly off Intelligence casting spells normally based off Charisma, which indicates a potentially different method of learning to cast the spell.

Of course, this makes the use of the same spellcasting DC for a tradition across class and archetypes questionable.