If you were using your PSN characters as the active save data for cross-save, you likely applied your season pass to those characters. That means you would not have access to the season pass for your Xbox characters, unless you separately purchased the season pass on your Xbox account since deactivating cross-save.
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I purchased Shadowkeep Deluxe Edtion 1st on my Xbox as I knew I would be using cross-save and I purchased Shadowkeep Deluxe Edition second for PSN. Sort of a bummer if I will not have any access to season pass content on one console. Though I suppose if I have access to everything (minus seasonal content on one console) I am okay with that and if need be I will access seasonal content on one console.
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If you had cross-save active, with your PSN characters as main, and purchased the season pass (via deluxe edition, or otherwise); then the pass was applied to your PSN characters. That is how the season pass setup works. If you already had cross-save active prior to the Shadowkeep launch, then there was no need to buy the deluxe edition twice. You only needed to buy it once, as the season passes would be applied to the active characters only, and then buy the standard edition for the other platform. If you were only going to be playing on one platform, not both, then you didn’t need to buy the second copy of Shadowkeep at all.