Reinforcing what @mwra says, I’ve found that I can use the same TB program and (single) TB license across my three computers, as long as I follow two rules, one trivial and one very important.
The trivial one is to enter the license number separately on each computer, when I renew my TB subscription each year. (You don’t need to renew annually to keep using the version you already have; you just need it to use subsequent releases and betas. Ie, if you want, you can pay for the program once and use it forever. For me, the ongoing stream of innovation is more than worth the annual fee.)
The more important rule is to make sure to have any specific .TBX file open on only one computer at a time. I use Dropbox as the storage site for the files, but the Dropbox/TB combo is not designed for real-time synchronization. You open the file on one computer, work for a while, and then you close and save that file before opening it on another machine. After Dropbox has had a chance to note that change, you can safely open the file on the next computer.