I don't believe there is an equivalent in the UNI designation system. First, let's clarify exactly what the grade is because there is no such thing as AISI 204Cu. AISI stopped maintaining product standards many years ago, which means that SAE and ASTM are the appropriate North American standards to reference. There is a low-Ni austenitic stainless steel with the UNS designation UNS S20400 made by AK Steel and marketed under the Nitronic 30 tradename. I am only aware of it being available in flat products (sheet, plate), and it is included in standards such as ASTM A 240 and A 666.
UNI EN 10088-1 is a list of all the official stainless steel grades referenced in the various product standards such as UNI EN 10088-2, UNI EN 10263-5, etc. I could not find any exact equivalent to UNS S20400, nor anything I would call a close equivalent. Grade 1.4597 and 1.4371 are similar, but not equivalent. If AK Steel is your material supplier then I would talk to them about this issue. Perhaps referencing an ASTM standard like A 666 would be sufficient.