I've seen the expression "neck tingle" used in the context of enjoying a good song. Is this a similar expression to "goose bumps"? If not, what would be the difference in use?
This sounds like the forming of a noun from a longer phrase, a relatively common practise in slang.
"It made (the hairs on the back of) my neck tingle" is a common phrase for something that is either exciting or frightful. This is sometimes expressed as an adjective or adverb phrase: neck-tingling or neck-tingly good, for example. Both of these are very much non-standard but might be used in sensationalist description, like music reviews.
From there, it's a fairly short step to "neck tingle" as a noun: 'It gave me a neck tingle', perhaps.
Another, much more established, example of this adaptation is 'bone-chilling' to describe something that chills the bones, either because it is scary or literally cold.