Is the reason I'm not able to see devices that are "passerbys" in my router that my router doesn't have that feature?
Yes. By router I am assuming you mean some sort of consumer gateway device with built in 802.11 wireless (hereafter just "wireless"). These devices will not have many of the features found in enterprise solutions.
Any wireless device has to be at least somewhat aware of the other wireless devices operating around it, if only to avoid collisions when transmitting.
How does a WAP see a passerby? Like is a mobile emitting something that the router can pick up?
When you want to connect to a wireless network on your device, it generally will provide you a list of nearby networks. There are two ways that a device can discover this information, one active and one passive, however the active probe request is typically more common. Most devices will check fairly frequently for networks in the area.
For more detail on those processes, you may want to check my answer to another question.
Aside from that, if your device is communicating on the same channel (even with a different wireless network), nearby devices will be able to detect and record their presence. A common misconception is that if you use an encrypted wireless network, this entirely protects you. However the truth is that only the data portion of the 802.11 frame is encrypted, the headers still need to be sent in the clear.
Lastly, is it possible for a device to hide itself from WAPs?
Possible, yes. Although, this is unlikely as it would require that your device make no transmissions at all. A device that doesn't transmit at all isn't very useful to most people.