I believe that is the case. I have an older version of InDesign and discovered that it can be installed up to three times (on three computers). Fortunately I could start the old computer and unregister the software... So when (before) you upgrade your computer it is a good idea to unregister the Adobe software. If you can't start the old computer, I assume you can contact Adobe and explain the situation and have them reset your "installation counter".
Adobe CC license can be activated on 2 machines using the same Adobe ID but only one at a time.
Please see Error: "The serial number is not valid for this product" | Adobe Creative Suite and Invalid serial number error for invalid serial number prompts.
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