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RAW, I don't think there's anything prohibiting different classes from selecting the same spirit, so I'd say RAW you would, but one pool would get +7 on the check and the other pool would only get +1.
Note that a Shaman is prohibited from selecting the same spirit twice, but there's no rules covering this for non-shaman archetypes.
RAI, I think it's really just an artifact of the archetypes and the Developers not thinking anyone would do what you propose. Me, I'd be inclined to generally rule that in all cases, you can't select the same spirit twice, whether that be a Shaman selecting his regular spirit and his wandering spirit, or a multi-class combo selecting spirits. It's not that I think it would be over-powered or anything, it just doesn't feel right to have two of the same spirit. The other possibility that I might consider is stacking based on the idea that you can only have one spirit of each type, so if you took the Lore Spirit as an Oracle 7 and as a Wizard 1, you only have ONE Lore Spirit which operates at Level 8. Haven't thought this through entirely, but that's seems like another viable option.
I would rule that they could not stack due to the Oracle spirit being based on Charisma and the Wizard Spirit being based on intelligence. With that you would get into rules where one would trump the other.
I'm leaning toward the same spirit can be selected by each class but cannot be stacked.
Unless there is language in the spirit rules or the rules for either archetype stating that they stack, they don't. Just as a paladin/cleric gets two separate pools of channel energy, you would get two spirits and separate spirit abilities. If the abilities are the same, they would not stack.