Tina,
I've created a test user to troubleshoot this issue.
I've created a new test communication site but this doesn't seem to change anything. When I select the invitation link The test site opens directly with no "star" present in the upper right hand corner (link to image below).
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When I select "Follow" in the email invitation I get same result - site opens with no star present.
It seems that the only way to have the star present is to go to the SharePoint home page where all sites are listed. When I do this a previously shared site shows up but not the new one. As you can see from the image below, I cannot get any sites to show
up in my "Following" section for some reason and the newly created site isn't even shown. A previously shared site, however, does show up and I've captured the message that pops up after selecting the star at this location (link to image below).
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As far as other users... I only have one test user that I've been testing SharePoint on. Permissions for this user were created in the new site's Advanced Permissions section > Grant Permissions Visitor. Test User had Read permissions.
Please let me know if there is more information that you require. Thanks for you help.
[PII is masked by the moderator]