Sentient, but constructed
According to the database entry concerning the Guardian of Forever at the official Star Trek web site:
Little is known about the Guardian of Forever. The Guardian served as a conduit for a race which traveled to a new age, thus leaving the planet – loosely referred to as the "Time Planet" – with the Guardian as its only indigenous, sentient resident. The Guardian is more than ten million years old. Its function seems to be that of showing the past of any planet, with the exception of the Time Planet itself, and of allowing journeys in time.
Not much is known about the civilization which built the Guardian, other than impressive, extensive ruins and the Guardian of Forever itself. The materials and principles on which the Guardian of Forever are based upon are completely unknown.
From this, we can conclude that the Guardian was constructed for a specific purpose, and so perhaps not alive in the traditional sense, but the entry affirms that it is nonetheless sentient, and so the parallel you draw with Soongian androids and the EMH is apt.
This is the extent of what is known in canon, as per the admission of the official web site itself.