Aaron is Hebrew and Erin is Irish. They just happen to sound the same.

They don't even sound the same, except in some (not all) American accents. They don't sound even remotely similar to me.

I thought they did not, but after actually vocalizing them I would say they are basically indistinguishable in real (fast) speech.

Once more I find myself wondering whether English really needs different vowels, if they can all be substituted for each other.

English is easy to speak but hard to write.

Interesting, they sound exactly the same to me, and I cannot even think of how they might sound different to someone else. Care to explain?

Exaggerated, Aaron sounds like "air-ron" and Erin like "ear-ron". It's similar to how some in the US pronounce "pin" and "pen" the same.

Not when pronounced correctly as A-Aron

To my Irish ear, these are two completely different names, not even slightly similar. (They're also etymologically unrelated.) It blew my mind when discussions on r/namenerds revealed that many Americans consider these names similar or even identical.

For most of my (male) life, people misspelled my name Aaron as Erin fairly regularly. I think that has contributed to the confusion. I also used to get Arron, Airon, and other variants. The "A-A-Ron" skit has made a noticeable difference in recent years.