Why would a Pokemon have the incense glow?

When a Pokemon has the incense glow, that means it was created/summoned by your incense. In turn, this means only you can see it, and any other nearby players will not be able to catch or view it.

In your example, if there was another player with you, they would be able to see the non-incense Pidgey, but the incense one would not exist for them.

This question and answer explains that incense only effects the user of them.

Based on personal experience, the incense floating around a Pokémon seems to indicate that it is a Pokémon that was spawned from using incense, whereas the other Pokémon spawned normally. I have noticed this when using an incense while tracking down a Pokémon that had already spawned.