Did ancient physicians feel that there were certain foods that might affect some people differently than others? Absolutely.

Are some of the foods that were fingered back then now known to cause food allergies? Absolutely.

Does this mean that "ancient physicians knew about food allergies"? Well ...

This article in the Journal of Allergy and Clinical Immunology (that's the journal of record for immunologists) goes over the historical record for food allergies. A lot of the evidence points towards milk and cheese:

From the texts of Hippocrates (460-377 BC): “Cheese does not harm all men alike; some can eat their fill of it without the slightest hurt…. Others come off badly. So the constitutions of these men differ, and the difference lies in the constituent of the body which is hostile to cheese and is roused and stirred to action under its influence. Those in whom a humor of such a kind is present in greater quantity, and with greater control over the body, naturally suffer more severely. But if cheese were bad for the human constitution without exception, it would have hurt all.”

But the article also includes admonitions against legumes, shrimp, chicken, meat, and nuts coming from a variety of cultures over a 1,500 year period.

The problem -- these explanations are completely devoid of the context on their time, and relying of a modern empirical context. Let's use the Hippocratic example example above, which comes from Ancient Medicine. I'm going to post the whole quotation (and it's from a different translation, so apologies there):

And it is not enough to know simply that cheese is a bad article of food, as disagreeing with whoever eats of it to satiety, but what sort of disturbance it creates, and wherefore, and with what principle in man it disagrees; for there are many other articles of food and drink naturally bad which affect man in a different manner. Thus, to illustrate my meaning by an example, undiluted wine drunk in large quantity renders a man feeble; and everybody seeing this knows that such is the power of wine, and the cause thereof; and we know, moreover, onwhat parts of a man's body it principally exerts its action; and I wish the same certainty to appear in other cases. For cheese (since we used it as an example) does not prove equally injurious to all men, for there are some who can take it to satiety without being hurt by it in the least, but, on the contrary, it is wonderful what strength it imparts to those it agrees with; but there are some who do not bear it well, their constitutions are different, and they differ in this respect, that what in their body is incompatible with cheese, is roused and put in commotion by such a thing; and those in whose bodies such a humor happens to prevail in greater quantity and intensity, are likely to suffer the more from it. But if the thing had been pernicious to of man, it would have hurt all.

The author in this case believes in a humoral model of health -- that balances and imbalances of body fluids cause all health and disease (and since he references Empedocles, it's probably safe to say they're referring to the four humors, blood, black bile, yellow bile, and phlegm). One of the chief ways to affect your humoral balance is through food. Still, no matter a man's constitution, there are some foods that will always prove harmful (like wine in large quantities, which is fairly uncontroversial). However, there are some constitutions (humoral balances) that can tolerate large amounts of a certain food, where others don't. Perhaps cheese helps a phlegmatic constitution, but hurts a melancholic. And the author also makes it clear that he is only using cheese as an example -- clearly there must be more foods that have this ability.

So I actually throw the question to you -- is this an awareness of food allergy? My gut feeling is no, especially when you compare it to early modern descriptions, such as asthma attacks after ingesting fish in a patient, in Oriatke, where exposure is clearly linked to either urticaria or anaphylaxis. But you let me know what you think!