As you’re talking about customers, I would assume there is money available, which means having options. I think you need to seriously consider whether you have a community or a collection of communities - and when you have decided that, ask yourself what makes sense in that context.
If you want each group to be truly segregated, a single Discourse install is not the way to go.
If I might comment on some specific points;
Is this a regular occurrence? It sounds like a hypothetical concern to me - I’m not sure why your customer’s users would be mentioning or messaging random strangers. I’m also surprised at users being notified of mentions if they don’t have access to the category in which it happened. That sounds like a bug, if you can confirm that it has indeed happened.
I do not wish to be unkind, but this is not a reasonable assumption.
You created a forum which has a community and members of that community are naturally discovering each other. I’m not sure this is the best possible implementation for your use-case, at least as far as I understand it. But you simply cannot expect the software to account for your atypical setup of an almost Reddit-like community of communities and radically adjust itself to accommodate your desire to hide this from people. You have a forum and it is acting as a forum.
If you want multiple forums (fora?) with distinct communities, then you can spin up additional installs of Discourse to fit your situation. I genuinely think this would be your best option in the long-term.
You might be able to find someone in Marketplace who can create a custom plugin to this for you. Please be aware that these people will be expecting to get paid for their work.
Would using display: none on certain elements not be sufficient here?
It’s not making the usernames completely impossible to discover, but we aren’t talking about the likes of people’s private health information. When users can already discover usernames by manually tagging people and seeing auto-complete suggestions, I would this would be sufficient…