In SAP MM, in Purchasing view there is a field called "Order Unit" (BSTME). Based on the technical info [attached] it sits in MARA table, however, when choosing the Purchasing view the UI asks to choose a plant. Is this field global and once set, it will be shown as the same unit in all of the plants, or we can choose different order units for different plants? If the latter, how is this possible since MARA does not have the plant as the key field?
Yes, it can be.
If you go to purchasing pricing conditions, you can find that one of the possible combinations is
67: Plant Info Record per Order Unit
So it depends on what key combination will be used in which case based on concrete order conditions, but this situation definitely can happen.
You can read more about purchasing info-records here:
https://blogs.sap.com/2013/12/27/create-info-record-for-different-order-unit/