Are Access Points Routers?

Quite simply, the answer is no, Access Points have no routing ability because they are not Layer 3 devices.  Sometimes people wonder why Access Points have IP addresses if they are not Layer 3 devices, the answer to that question is the IP address is there only for management purposes.

In their most basic form, an Access Point (AP) is really nothing more than a media converter.  The AP takes Wi-Fi frames that are absorbed by it's antennas and removes the payload and then encapsulates the payload in an Ethernet frame and transmits that frame on the wired network.

It's true that cheap consumer grade APs have primitive routing functionality, commercial grade APs typically do not because of their role in the access layer of the network, whereas routers are associated with the distribution layer of the network.

Access Points are layer 2 devices, much like switches.  The interfaces of an Access Point have MAC addresses not IP addresses and their interfaces are in the same IP network or subnet, something a router cannot do.  Access Points function as Bridges, bridging traffic across interfaces.

When you ping your gateway (the interface up the next upstream router) from your Wi-Fi enabled laptop, your packet passes through the AP unchanged as it travels to and from the gateway.  At layer 3, Access Points are transparent.

Routers route traffic between networks and they exchange network reachability information with other routers.  Access Points when in a Mesh topology, use a Spanning Tree Algorithm to eliminate bridging loops and find the optimum path through their layer 2 network.