I wonder if "allocation" might be a better term to use --> "Alloc."

I'd say "Assgt." should do fine

I would prefer "Assgmt.", to be even clearer

To my knowledge, there is no standard abbreviation, which is one reason why you're getting different answers from everybody. But there's one intriguing possibility, depending on the meaning of 'assignment' here: if you mean it in the sense of 'task' like 'my homework assignment', then this is not applicable. But if you mean it in the sense of an assignment statement, (e.g., x = 3) then you could consider using a left arrow, which in the context of software-savvy people would be understood to mean assignment, more so if the table had some text above it where they talked about assignment. What about labeling your column like this: 'x <- ?' or 'x = ?' ? This wouldn't work if the table is destined for consumption by the general public, but for programmers, it would be ideal.

Good to know that I'm not the only one struggling with the differentation between 'assignment' and 'allocation' (including reasonable abbreviations). So I take the chance to get approval/disapproval on how I used these words.
'You have to assign a specific supplier to the purchase requisition' and 'The cost are allocated to the general expense account'or the goods are allocated to customer xyz. In other words: When talking about 'allocation' some kind of a reservation is implied. In case of 'assignemt', however, I create some kind of a relation. Can you agree on that?? Hope so.

I forgot to give context for my query. The "Zuord." thing appears in quite a few columns, sometimes alone, sometimes with something like "Zuord. Ext. Sprechtaste". It concerns a remote alarm system where an emergency button can be pressed, it triggers an alarm in a control center and the doctor/ambulance is sent to the person immediately, since his/her address, previous medical data etc. is saved in the system. In this case it is mostly a phone number that is being assigned to a name/unit/button (if the button is pressed, that number is dialled, or a phone number is assigned to a name in an address list, etc. I am not very sure about the assigned/allocated thing either...

UMW: About the assigned/allocated thing, I think that you gave a good description of the difference. You could assign a supplier to the order, only to find that that supplier was not available, refused the order, or whatever. If you allocated a supplier to the order, it sounds like you have the power to enforce it (because the supplier works for you, or some other reason).

Ich würde in dem Fall tatsächlich 'alloc' nehmen, da die Aktionskette ganz genau spezifiziert ist....