The defining features are transparency and regulation: an exchange must display its quotes, publish its rulebook, apply price-time-priority or another stated matching rule, and register with the regulator as a self-regulatory organisation.
That is what separates it from an alternative-trading-system or a dark-pool, which match orders privately and report trades afterwards.
Example: a limit order resting on an exchange at 25.00 becomes part of the nbbo and is a protected-quote that other venues may not trade through. The identical order resting in a dark pool is invisible and protects nothing. Exchange groups in the US operate multiple separately branded venues with different fee models.
Related: alternative-trading-system, lit-market