Why does this Garmin GMA340 have an Airway/Inner marker indicator? On this Garmin GMA 340:


the outer marker is blue and is designated O
the middle mark is amber and is designated M
the inner marker is white and is designated A and placed on the far left.


I looked at the GMA 340 Manual and could not find an answer although it does say Airway/Inner Marker.

Why would it say Airway if this is what the A indeed stands for and why even have an inner marker because that Garmin can't do a Category II approach?
The "A" is not an inner marker. The A is an airway marker. Airway markers were part of a now-obsolete system of "red" airways. They were placed along the enroute path and they could be picked up at cruise altitudes. The red airway system was mostly abandoned in the 60's and few "A" markers are still operational. Here's a map showing where they were located. They weren't at aerodromes. 

Source

The airway markers transmit the same 3000 Hz audio frequency as inner markers so the airway marker can be used to pick up the inner marker. Since you say that unit is not certified for cat II it makes sense that they labeled it "A" instead of "I" although it can be used for inner markers.