Your description is still too vague to give a reasonable response.
Same dimensions, even weight is the same. For example, someone puts 3 trays at a table, and he forgets to take it away. The next person comes with another 3 trays and the trays are mixed. So if the previous person came and wanted to retrieve the first 3 trays, how does he know that the bottom ones are the ones that hes looking for? We can use labeling to easily solve the problem, but is there any other ways such as using sensors to solve this kind of problem?
If the 2nd person has total freedom to move the trays and rearrange them in any way they choose, then there is no way to tell the difference between identical trays, unless you want to get into some highly detailed examination of the trays, such as recording exact patterns of dents and scratches on each tray, or having the trays tested for fingerprints or residue left on the trays.
How are you currently telling the trays apart? There has to be something that differentiates them, or you wouldn't yourself be able to tell which set was which.
Labeling would work, or using trays with an embedded RFID tag or similar identification technology, but without more knowledge of how these trays are being used its hard to suggest a method.