Doesn't referring to a specific individual (somewhat) defeat the purpose of the order?
Addressing people by profession means they are being recognised for something.
The question we need to ask is, at what times is it hardest to achieve something without names?
So here are my answers and potential solutions. Note that in some cases, things are going to end up being less efficient, but the monks believe it's a worthwhile trade-off.
1. Getting someone's attention.
Let's say in a room with several monks, Brother A sees Brother B with a toolbox and wants to ask whether he has a chisel. Or, Brother B seems to be carrying something to the wrong place. Brother A would refer to Brother B by his location or activity (e.g. Brother with the toolbox, etc.) The reference is based on the immediate situation, rather than the person's identity.
Bob can speak Latin. Bob can carve wood expertly. Bob remembers which of the hundred shelves a particular scroll is on.
Well, too bad. The monks gather several times a day for several reasons, one being that someone can ask the whole congregation for someone with such skills/experience/knowledge.
3. Division of responsibility. Scheduling.
Building a chapel requires someone to come up with a plan, someone to co-ordinate the work, and many to perform parts of the labour. Too many cooks spoil the broth and so forth. Similarly, if something must be done in shifts (watching the flame, ringing bells) then some system is needed for scheduling who is where at what time.
In this case, at meetings people are asked to volunteer for jobs/roles, and they raise their hands to claim the task, and then know it is their duty to show up and perform it. Nobody else need know who they are. Alternatively, they can place black marks on a roster or responsibility chart and, in cases where needed, be present at a meeting place at particular times so that others can co-ordinate with them without knowing who they are.
This one is not so much a question of names but more an ongoing teacher-student relationship which implies recognising someone and gaining respect for them. While it may be seen as a "necessary evil", an alternative is to intentionally rotate teachers and students, and encourage a "seeker/scholar" approach to learning whereby a student pursues knowledge and skills so it really doesn't matter if they consult a different person each day; indeed perhaps it is quite beneficial. In addition, there can be an emphasis on "teachers" writing down their knowledge anonymously and "students" reading it.
5. Referring to third parties.
Brother X is missing! Is he ill? We need someone to fill his place! Well, that's what the frequent meetings are for. If Brother X is ill, he can let someone know what his task was and that person will announce it without having to name Brother X.
Or let's say Brother Y doesn't realise he's good/bad at something. Can you warn others? No. You just have to tell him, and it's his responsibility to act accordingly. That's how the order works.
Brother A knows that B possesses an item that brother C needs. But A cannot identify B to C. A can simply tell C that someone has the item and during one of the meetings it can be requested. There are no secrets in the order; after all, they would be a selfish thing.