It's original meaning was days since software release, without any security connotation attached. It came from the warez scene, where groups competed to crack software and make it available to the scene earlier and earlier. A week after general release, three days, same-day. The ultimate was 0-day software, software which was not yet available to the general public.

In a security context, it has come to mean days since a mitigation was released. Prior to disclosure or mitigation, all vulnerabilities are "0-day", which may be for weeks, months, or years.

It's not really an inflation of the term, just a shifting of context. "Days since software was released" -> "Days since a mitigation for a given vulnerability was released".