I may be wrong, but it reminds me of sth called a 'health card' being needed in my country (.pl) when you are working with food. If it's true, it only proves their legal system concerning sanitary measures has way to broad categories and somehow olive-oil (food-based) falls into 'food' while running an online store falls into 'having a place where you deal with food'. Providing stool samples of shareholders is just an extreme example what happens when laws are too general and not in time with modern business models and current technology. It's easy to figure out that broad laws open to interpretation are easy to abuse by corrupted officials.

I don't know for a fact, but I doubt that this specific detail is enshrined in law. Much more likely is that the law requires certification from a bureaucrat that you are not carrying some terrible disease.

Greek bureaucrats have a lot of leeway and very little accountability, and their positions are openly coveted (among a portion of the population) for ability to extract bribes (it's very desirable to be a tax assessor, for example). These requirements may just be the bureaucrat's opening offer in the bribery negotiation, which can then be made "unnecessary for an upstanding citizen such as yourself".

It sounds too absurd to be true, but they are selling olive-oil and olive-oil derived foods and cosmetics.

Anything food-related, cosmetic-related, or even just olive-oil related is likely to have some arcane regulations, especially in the Mediterranean. (I've occasionally heard the olive oil industry being associated with cartels and frauds.) That goes for double if they are – or get confused with – a manufacturing facility which formulates its own products onsite.