Is SharePoint in Azure the same as Office 365?

No. SharePoint in Azure is not software‑as‑a‑service. It would be a SharePoint installation specific to your organisation that would be deployed on Azure's infrastructure‑as‑a‑service. Office 365 is software‑as‑a‑service and is a multi‑tenant deployment where Microsoft is responsible for updates and migrations to new versions. There are pros and cons for each platform.
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