Use another Macbook as a extended display for a Macbook.

I would guess the limitation here is with the older Mac (the MacBook Pro mystery Mac) and its (lack of) AirPlay 2 capabilities, and not with the newer Mac (the MacBook Air M1, 2020).

What model MacBook Pro? MacBook Pro introduced in 2018 or later? That should work. But if I’ve guessed correctly here and yours is an older MacBook Pro model, then not so much. Unfortunately, Apple and Intel have had a various Core i5 models, which makes that particular detail ~useless for distinguishing MacBook Pro models. Given it’s an i5 and still on macOS 13, and given it works with Universal Control, it’s quite possibly a MacBook Pro 2017 model, which won’t work on macOS 14, and won’t work for AirPlay to Mac, but will work with Universal Control.

“Linked keyboard and mouse” is Universal Control:

… Universal Control: Use a single keyboard and mouse between Mac and iPad - Apple Support

Extended display is AirPlay to Mac (AirPlay 2)

… System requirements for Continuity features on Apple devices - Apple Support

… Set up your Mac to be an AirPlay Receiver - Apple Support

And more for completeness as it too arises in these contexts, Sidecar uses iPad:

… Use iPad as a second display - Apple Support

As for your “Why?” question, we often don’t know those. Only Apple. In this case, I’d guess that there were hardware-related changes needed to add support for AirPlay to Mac.

To add to the excellent answer by MrHoffman:

AirPlay v1, and AirPlay from a Mac to an Apple TV set-top box have been around for a very long time. AirPlay v2, and AirPlay to a Mac, are much more recent. This is why things are not symmetrical.

Timeline:

• In the beginning (2004), was AirTunes, an iTunes feature for streaming music to an AirPort Express.
• AirPlay v1 appeared in 2010 – 2011, for streaming audio and video to Apple TV set-top boxes.
• AirPlay v2 appeared in 2018, with improvements to buffering and for streaming to multiple devices.
• AirPlay to Mac appeared with the release of Monterey in late 2021.

https://en.wikipedia.org/wiki/AirPlay

Notes:

• Several 24" and 27" 2009 and 2010 iMacs supported Target Display Mode, using DisplayPort input. It seems very likely that the design center for AirPlay 1 was "connecting an iPhone to a HDTV", not "allowing the use of another Mac as a display,"
• In 2011, Thunderbolt 1 came out. This was good for people wanting to add docks and high-speed drives, but made Target Display Mode less useful. TDM now required Thunderbolt input. Initially, there would have been very few Macs or PCs able to provide it. These days, you'd need $80 worth of adapter + cabling hardware to make a TB 3/4 to TB 1/2 connection (and in most cases, you'd then run into restrictions on "the other Mac").
• Target Display Mode went away when the first 27" 5K Retina iMac came out in Late 2014. It never came back.
• By the time that AirPlay to Mac appeared, Target Display Mode had been dead for years.

You cannot use one Mac as an extended display for another Mac, period.

The feature you are attempting to use only allows you to seamlessly move pointing device and use keyboard across two different Macs to be able to use both Macs at once with a single keyboard and pointing device. It does not provide using one Mac as an extended screen.

I have the same problem did you solve it by chance?