What would be considered thick milk in old recipe
I am pretty sure she means milk which went sour naturally, or spoiled milk. 

Before refrigeration and even after refrigeration but before ESL and UHT milks, a milk that got sufficient microbial activity would visibly curdle and get thicker and somewhat sour. Such milk is unpleasant to drink, but people used to not want to afford to throw this milk away, and took the risk of using it within something that gets baked. The reasons for baking it are 1) to hide the taste, and 2) to reduce the risk of food borne disease. 

My own family has this kind of recipes, and I still remember my mother using them in this way when I was a child (it was in the 90s, but food safety rules hadn't reached us yet). Also, this is how names like the German Dickmilch (which literally translates as "thick milk") came to be. So I think it's plausible that this must be the case here, given that your recipe is old. Also, if the recipe really has no flour, using soured milk improves your chances of getting something firm and custardlike (albeit grainy), as opposed to the much more finicky flans made with fresh milk. Although I must admit my mothers' recipes wouldn't be for a completely flourless cake - she had some that were used as filling for baked pastries, and a savory one which contains just a little bit of flour, gets poured into a pan and baked. It doesn't form a doughy texture, but rather something like a pie filling, or a quiche without crust and vegetables. 

For recreating the recipe nowadays, I would suggest that you use the stuff that American supermarkets sell today as "buttermilk". This is milk soured by known-harmless bacterial cultures, somewhat similar to yogurt. If you live in a part of the world which uses the term "buttermilk" in its original meaning (the watery liquid that is left after butter has been beaten out of the current-american-meaning-buttermilk) try searching for similar cultured products under a different name, maybe "cultured milk" or "soured milk". If you cannot find any, appropriate substitutes would be kefir or yogurt thinned with milk, even creme fraiche thinned with milk for a creamier result. I am not a fan of adding acids to fresh milk as a substitute, in my experience this creates the wrong texture and it is difficult to get the acidity right. Do not let your own raw milk out, this is unsafe by modern standards, and our grandmothers were wrong in assuming that it will become safe by heating.