This is a known limitation of items that boost spell attack bonuses and spell save DCs; the system cannot currently distinguish between those attributes when multiple are granted by different classes due to multiclassing. As such, all such attributes will be boosted.
I have multiclass bard/artificer. When I carry and attune to all purpose tool, Spell Attack and Save DC for both classes are bumped by one.
I would only expect the artificer stats would be bumped.
This is a known limitation of items that boost spell attack bonuses and spell save DCs; the system cannot currently distinguish between those attributes when multiple are granted by different classes due to multiclassing. As such, all such attributes will be boosted.
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