I realize it can go bad after a long time, but why does it take so much longer than other dairy products? Does it have to do with the alcohol content? If so, what is the process that is happening?
Here is a quote from the company that makes the registered trademark Baileys Irish Cream:
The alcohol content is the sole preservative listed.
Ingredients: liqueur, acidity regulator (E331), emulsifier (E471), flavours (caffeine), colouring (caramel (E150b).
Source: a food product inventory database
The alcohol keeps the product from microbiological spoilage, the E331 (sodium Citrate) buffers the product form getting damaged by acids produced by any spoilage, and E471 (mono and diglycerides of fatty acids) keeps the fat from the cream stable in Emulsion.
Source: my opinion as a food scientist