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Actually this issue is related to Magento full page or block cache. But with fresh magento it should work properly so issue is related to custom module which used for customer login or session.
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This is the expected behavior from 2.3.4-p2. We can see this has a new update from 2.3.5 release notes
https://devdocs.magento.com/guides/v2.3/release-notes/release-notes-2-3-5-open-source.html#customer
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