Is there anything has been changed before the issue occurs?
This window "Select the credentials you want to use to logon to this SharePoint site" is shown when several authentication providers are configured for the same web application.
1.Please check which are configured for your web application.
Central Administration -> Manage web applications -> Select the web application -> Authentication provider.
2.You could try to change the authentication provider in Central Administration for each application to NTLM.
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