How to tag distinct options/entities without giving any an implicit priority or suggested order?
An example of the problem in an aggravated form surrounds the controversy of France changing ‘mother’ and ‘father’ to ‘parent 1’ and ‘parent 2’ in official paperwork - where the controversy suggests the new standard implies one parent is 'secondary' and the designation may induce completely unnecessary family conflicts.
In technical writing this may happen also; we have two or more completely independent identical units/objects/devices, which need to communicate. Any of them may initiate the communication, and this will assign them specific roles, but before the conditions occur, they are perfectly equivalent and so suggesting any order, priority, sequence etc would be misguiding - but we still need to distinguish them; assign them some designations when describing the situation. Marking them "Unit A, B, C"; "1, 2, 3"; "X, Y, Z", "Alpha, Beta, Gamma" this all is a specific sequence. I might try using symbols, 'unit @, unit *, unit %' but I believe this by itself would be rather confusing, never mind not yielding itself for verbal communication.
Can you suggest a convenient set/system of identifiers to use e.g. in technical writing or legal documents, that doesn't imply any order or priority of the options, but still allows to reference them uniquely?
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Full names and arbitrary names are good solutions to the question you asked. To address the question behind the one you asked -- the implicit "superiority" in ordering -- write examples that don't start with the first unit. For example, I might describe a database with nodes A, B, and C, and then talk through an example where B acts as the initiator in processing a query. Who says it has to be A? The names are arbitrary, after all, so don't start all your examples with the first name in your ordered set. (For that matter, why not have nodes K, L, and M?) If you have users Alice and Bob and Carol and Dan, try having Dan or Carol be the first ones to act in a scenario.
There is value in having sequential names in some kinds of diagrams and examples, like that database cluster (where there might be way more than three nodes). Don't make your documentation less usable by talking about nodes 12, 37, 42, and 139 instead of 1-4 or A-D. But you don't always need meaningful names and you don't always need to match "first in the sequence" with "first in the example or sequence of actions".
Another approach is to use words based on function. For example, a tax form I fill out every year has places to list two names (for joint returns), "filer" and "spouse". In the database documentation that I work on, we sometimes refer to the "initiator node" -- which could be any node in the cluster, but in the context of a particular operation, it's the one "driving" the operation.
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I think you may be overthinking the issue.
In technical writing when you name three entities with elements of a specific subset, the ordering of the specific subset doesn't come into play unless it is specifically stated. There are plenty of examples where the common "A,B,C", or "X,Y,Z" are used without underlying assumptions of "who come firsts" or "who is more important". Luckily enough, technical writing is somewhat shielded from those kind of controversies.
Answering your question, though, you could try:
Assign full names to your entities. This is often done in telecommunications examples or in cryptography (Alice and Bob, exchanging messages...). If you don't like inventing name, you could use the Nato Phonetic alphabet. To be sure, following an alphabetic convention won't free you of an underlying order. Another drawback of this soluton is that full names are not concise; if you have a lot of entities to name, you'll see your text fill up with Alices and Bobs.
Use a color coding. Your entities can become Red, Green and Blue. This is somewhat assimilable to the alphabet, since you can easily shorten those to RGB. Yet, if you pick your names from colours, nobody will be able to claim that you are making assumptions about who's more important.
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