Where do Individual Traits start? The distinguishability of organisms is dependent on their complexity, especially in near complexity. For almost any organism with any sort of ability for memory, there should be some distinguishability. As behavioural characteristics are mostly due to life experiences. For more simple nervous systems, ones without memory or cognition, there still can be noticeable differences, but not due to life experiences, but rather to due slight differences in development, either due to genetics, or early life development environment itself. As long as an organism has behaviour, (a coordinated response due to stimuli) there may be some ways to differentiate. It may become difficult at the level of bacteria for example, as mutations are so few between generations, that there may be two organisms that are genetically identical. In these cases the only hope you may have in differentiation is in the tiniest individualities in cell structure, causing slightly different reactions to stimuli.
So truly, as long as two organisms aren't completely identical in every way, given enough time, you could theoretically distinguish between organisms.