In reply to Olaf Prot:
From a quick Google, it doesn't seem like you can do this, unless you're able to "link" one account to the other, which isn't allowed between two personal accounts or two business accounts. And even then it only seems to allow quick switching rather than having both connected simultaneously, and wouldn't scale beyond two accounts if you end up with more clients.
If you're working with multiple clients, would it be feasible to instead have a single Dropbox account of your own and get each company to share relevant folders with you?
The other option people suggest is to have separate Mac accounts for the separate businesses. But that could be clunky if you switch back and forth a lot. If you're reasonably geeky, you could maybe experiment with running Dropbox in separate VMs or Docker containers for each client then mounting the relevant folder.