Currently, Nintendo seems to be applied randomly to images of any (but not all) of their properties.

I'm personally refraining from adding the tag unless it's a crossover picture.
Should we have it for all these images though (implicating Nintendo properties -> Nintendo ?)
Or should it be only for crossover/official images?

Should use the same policy applied to the Type-Moon tag as you're suggesting.

I think going from most discussions we've had on company tags like this in the past (for one example, topic #6634), they are only to be used in specific cases such as crossovers, a company mascot without a particular copyright, the company's logo appearing, etc.

Roughly half of the posts tagged with Nintendo are deleted shit posts. That's not fun.