Can Dropbox even allow different devices to have the same device name?
Or is Dropbox sensing some other unique identification beyond a physical device like a particular version of Windows or install date?
Authentication mechanism OAuth2 is in use for Dropbox. Dropbox don't really care too much of your device name or anything else similar. Any such information collected is used for display purposes for your convenience, nothing more.
About installing/accessing problem, different things could lead to such effect. You don't mention if some device count restriction apply to your subscription (Basic plan only in fact). If there is, take a look on your currently connected devices and if need make decision what activity on is most appropriate. Another possible reason is an initialization problem. In such a case would be fine to Unlink your account from the application Preferences, Account tab and try reinstall earlier version (this one for example). Choose the offline installer! Before reinstall make sure "Early releases" on the general settings is turned off.
Hope this helps.
Do you work or study in the creative industry? If so, which area?