Requirement to buy an extra licence to move DT3 from one Mac to another?

You need to go to your account (login), where you will need to go to the section headed Your Licences, select DEVONthink, and then deselect your previous Mac under Devices/Edit, freeing up a seat on your license which you can then use for the new Mac.

Alternatively, if you still have access to DEVONthink on your old Mac, open it and select DEVONthink > License… > Remove, which will also free up your license for you to enter it in the same location on your new Mac.

You might find this post helpful if you want to migrate preferences etc. to the new Mac, if you have not already done so (e.g. using the macOS migration assistant, which I think I read here somewhere does the job for you).