iPhone 13 not recognizing older gen Apple tv

Verify that AirPlay is (still) turned on, on the Apple TV box.

Verify that your router bridges the 2.4 GHz, 5 GHz Wi-Fi bands and wired ethernet internally into one logical network. Else devices on one segment may not be able to communicate with devices on another segment. It is often enough to not use separate names for the 2.4 GHz and 5 GHz Wi-Fi bands. Recommended settings for Wi-Fi routers and access points - Apple Support

Verify that your Apple TV has been/is still activated properly. If that is not the case, then it would say so in Settings﹥General﹥About﹥Activation﹥Failed. Such would prevent AirPlay functionality.

For screen mirroring, both devices needs to be on the same (home) network. If your iPhone is on cellular but not Wi-Fi, then it may not work.

For screen mirroring, AirPlay must be activated (turned on) on the receiving device. Settings﹥AirPlay﹥.

Try restarting all the devices involved, including the router.

Use AirPlay to stream video or mirror the screen of your iPhone, iPad, or iPod touch - Apple Support

If screen mirroring or streaming isn't working on your AirPlay-compatible device - Apple Support

Not all content can be mirrored. Some content may be blocked by the content provider for mirroring. Some content may require a more modern receiving device.

older gen Apple tv

Identify your Apple TV model - Apple Support

I’m connecting it through that [Micro-USB] port … Am I supposed to be connecting it through hdmi or Ethernet or something?

Use the Micro-USB port for this. It is for “service and support” [1] [2].

Use a generic USB cable that supports data transfer as well as power. USB cables from other devices may be device-specific or charging-only.

Apple TV (2nd generation) should get enough power through the USB cable.

Apple TV (3rd generation) should be powered through its own AC power cord.

recognizing

Recognizing for what? Are you attempting to AirPlay? Are you attempting to use your iPhone’s software remote? Or something else?

it does say activation > failed

Please see here: discussions.apple.com/thread/253315520

how other users solved that error.

Else contact the Apple staff about this error message. [1] [2]