The white part is curd and the yellow part is whey. They can be separated in milk by the presence of acid, and helped along by enzymatic action. More acid typically yields better separation. Fresh cheeses are produced by directly adding acid such as vinegar or lemon juice. In cultured diary products such as yogurt and all other cheese, the acid is produced by bacteria like Lactobacillus. The reason fermentation acts as a short-term preservative is that many harmful pathogens do not fare well in an acidic environment, and when fermentation proceeds properly the "good" bacteria will simply outcompete the "bad."
When milk starts to "go off," that's because it's actually going off in the sense that most other people understand the term: it's starting to go bad. Whatever small amount of bacteria was in the milk has started to multiply, and has started to lower the pH. Coffee is not acidic enough to curdle fresh milk, but if the milk itself is slightly acidified then coffee will curdle it.
The fact that your yogurt separated so thoroughly is a sign that some acid-forming bacterium did serious work on your milk. I like Siggi's but I'm not a microbiologist, so I can't say whether you just had a weird batch of yogurt or there was already something living in your milk. It's probable that whatever turned your yogurt into cheese is in fact harmless. But it's also not improbable that something very unpleasant grew alongside it.
I think you got lucky. This is probably how our ancestors figured out how make different cheeses, but it's not exactly a reliable method. It doesn't help that commercial milk production is potentially a less sanitary process than it might have been in the past (although this might depend on historical place and time), and that the bacteria living in your refrigerator are probably very different from the bacteria living in animals stomachs and wherever else people used to store their milk in prehistoric times. I don't know if they had listeria back then, but we have it now and it is something you absolutely do not want inside you.
Throw it out. Unless you know any microbiologist friends, in which case you should bring it to their lab have fun seeing what's in it.