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.

The uniqueness of an individual's behaviour goes far beyond the present capacity of the human intellect. Consider that the human being does not even have the capacity to understand the unique behaviour of a basic inanimate particle like a photon, or an electron. Since the activities of these basic particles have a great affect on the activities of larger things, it is very obvious that human beings do not have the capacity to understand the uniqueness of an individual being's behaviour.

Some bees are just lazy. This is down to feckless parents and too much socialism meaning they don't have to fend for themselves.

https://elifesciences.org/articles/62850

Some ants are fussy about their nests whilst others live like slobs. This is because their parents never made their them tidy their bedrooms. It's got a lot worse over recent years because of a spoiled generation bringing up ants who've never even heard the word 'no'. Trust me. It's all over facebook.

https://www.newscientist.com/article/mg23331114-200-personality-of-individual-ants-helps-colonies-choose-best-nest/

Some bees are just lazy. — Cuthbert

Those are the drones. They are born to be lazy. But they have perhaps the most important job, to fertilize the queen. I guess they are given the privilege of being lazy, to ensure that they willingly carry out the one very important job that they do have.

So truly, as long as two organisms aren't completely identical in every way, given enough time, you could theoretically distinguish between organisms. — Louis

This does not even require organisms as it is also true of any two bricks in a wall or in the extreme case of any two electrons that don't even have substance. Bricks can be distinguished by testing but electrons can't. They are equivalent for most purposes but never identical in every way.