The short version is no, DTM does not expose the rule name for you to reference.
I have spent a fair amount of time trying to find some unofficial method or property with it, but it is simply not exposed within the scope of anything you can passively do.
I have also spent a fair amount of time figuring out the least invasive way to alter the _satellite object to give me what I want, and.. while I have done it in a sandbox, it is not something I have or will ever actually use in practice. (Adobe won't even support passively using existing internal methods/properties, let alone actually changing the DTM library).
I'm not aware of a way to accomplish what you're looking for as it will likely require a bit of custom coding. One of our super users on the community ofter comes up with solutions for use cases like this. I'll tag him below to see if he has any ideas.
for this to work, the condition needs to know the name of the rule/context it is firing within.
The reason for this is that if the user then does and interaction on the page (without reloading the page) that would make the "[custom condition logic]" return true, we can iterate over the rules that previous failed and trigger them off.
This is all related to the General Data Protection Regulation (GDRP) changes
If you are writing a custom condition within a Page Load or Event rule, that condition is only applied to that rule. Are you trying to debug something on the page to figure out which rule the custom condition is in, or is there another use case I'm not seeing?