I haven't had a chance to take the shinkansen as I toured Japan, but I did take a similar one in China from HangZhou to Shanghai more than 10 years ago. I heard that particular route was too short to reach top speed.
Back to the question, this is a hard question. There's the japanese people's "common" perspective on their "common" references and naming, and then there's the mathematical sets perspective.
I'll leave the Japanese perspective to someone who lives in japan to answer that to give further confirmation. Here is my guess at common usage terminology of the two terms:
Generally, the electric trains (don't think diesel/petrol trains can run underground without proper ventilation) if it goes underground like a subway, then technically chikatetsu 地下鉄 is correct.
The 新幹線 shinkansen refers to the bullet train, and usually for the purpose of differentiating with other train services in conversations, the term may be referenced to mean the more premium service train with premium pricing and probably long-haul routes. It may also bring a connotation that someone is traveling to somewhere far away, far out of the range of common electric trains. Therefore shinkansen might not be commonly referred as the subway as the latter might be cheaper and run shorter routes between stations or in a smaller area. My guess is that if both the shinkansen running underground exists in an area together with the subway, then speakers who want to be understood will choose the terms to clearly disambiguate one from the other.
For mathematical sets: If we group "trains that travel underground" to contain subway 地下鉄 chikatetsu, then the shinkansen, which is technically a train, and which may travel at certain segments of its service routes going underground, then at those underground routes it may be grouped together in the same set.
Here's a translated article a shinkansen route built 40metres underground. Can't find the NHK link though.
