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I noticed whenever I get a photogrammetry dataset, the products (like DTM, DSM) are always misaligned from satellite imagery of the same area by a few metres. Today I received a LiDAR dataset and they were perfectly aligned with satellite imagery I have of the same site.

Is there something about processing point clouds and related products that is different between the two, that would cause LiDAR data to be more geographically accurate/precise?

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No, the technical differences between Lidar or photogrammetrically-sourced point clouds have no influence on the final accuracy of datasets. Either can be off or "perfect" depending on the work done and the care of the creators.

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