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Based on how Purview enforcement typically works, this difference between Desktop Word and Word Online is usually related to where enforcement is happening.
Desktop Word uses the Office client + endpoint DLP integration, so blocking clipboard actions works there.
Word Online relies on service-side DLP (SharePoint/OneDrive) and endpoint/browser integration. The Edge extension alone doesn’t automatically enforce clipboard blocking unless Endpoint DLP and browser-based protections are fully configured.
I would check:
Whether the device is onboarded to Endpoint DLP
Whether the DLP policy targets endpoint devices (not just Exchange/SharePoint)
Whether browser protection settings are enabled
Whether the sensitivity label includes protection vs just classification
If desktop works but browser does not, it usually indicates enforcement is happening at the Office client layer but not the endpoint/browser layer.
You may find this helpful: https://learn.microsoft.com/microsoft-365/compliance/endpoint-dlp