The answer to the second part of your question if Anycast Gateway can be implemented without VXLAN/EVPN, was given by the guy who (quite literally) "wrote the book", Lukas Krattiger, here:
https://learningnetwork.cisco.com/thread/122986#661288
From which (full quote of Lukas' answer, highlight by me)
if there are two VPC domains and you use HSRP on all the four (same
VIP, same Group, same Password), there will be only ONE HSRP node
becoming active, one becoming standby and two listen. On the VPC
domain where you have HSRP active, you will see the first-hop
forwarding and ARP response to happen while the traffic from behind
the VPC domain with non-active HSRP will trombone to-be-route traffic
towards the VPC domain that has the HSRP active.
In case you change the HSRP Group or Password between the different
VPC domain, you get multiple HSRP active and as a result, the Switch
will detect duplicate MAC/IP situation sourced by the SVI that has
HSRP active (multiple). In addition, every ARP broadcast towards the
VIP will result in duplicate response as you have two HSRP active.
In short, HSRP doesn't provide you a clean 4-way active but if you
don't care on duplicates, log messages and other forwarding
deficiencies, you can hack VPC and HSRP to do an all active.
To your answer "can I do distributed IP anycast gateway without VXLAN EVPN", the answer is NO. You always have to configure the
VXLAN portion to make the distributed IP anycast gateway work. If you
interconnect to VPC domains back-to-back where each has distributed IP
anycast gateway, this would end it similar problem as above. Simple
solution is don't use back-to-back VPC and VXLAN EVPN instead and use
distributed IP anycast on all the host-facing nodes. We call this a
small VXLAN fabric :-)
And I believe the other answers in that thread also help with the first part of your question how Anycast Gateways work:
With the distributed IP anycast gateway, the default gateway is moved
closer to the endpoint--specifically to the leaf where each endpoint
is physically attached. The anycast gateway is active on each edge
device/VTEP across the network fabric, eliminating the requirement to
have traditional hello protocols/packets across the network fabric.
Consequently, the same gateway for a subnet can be used concurrently
at multiple leafs, as needed, without the requirement for any
FHRP-like protocols. (Building Data Centers with VXLAN BGP EVPN: A
Cisco NX-OS Perspective, at page 66.)
And an extension to that from Lukas, (again, highlighting by me)
to add some additional comments
the SVI representing the distributed IP anycast gateway announces its
IP and MAC address only towards the host facing Ethernet ports but
not towards the fabric (Overlay). As a result, the MAC/IP for the
distributed IP anycast gateway is only seen and reachable from the
host facing side and never from the fabric (Overlay) facing side. This
is rather important as there is no duplicate MAC/IP in the overall
network topology for the first-hop gateway and also ARP handling is
very deterministic (host facing).
In traditional Ethernet networks, such a host vs. fabric facing
topology determination doesn't exists. The challenges around this
missing awareness has been addressed through FHRPs that deals with for
example the ARP handling. In order to ensure only one of all FHRP
nodes is responding to an ARP request, you need to define a "master"
and this is where the FHRP hellos and election are coming into the
game.
hope that adds some clarity