Thunderbolt 2 device with TB3 adapter not recognized on Mac Studio

If using the rear ports does not help...?

Then it sounds like the older M1 Mac mini's clone does not have the correct firmware and/or software for the later Mac Studio.

I think that you need to Erase and Reinstall the Mac Studios, as per > How to reinstall macOS - Apple Support and then migrate your data from the M1 Mac mini or its backup.

Is this a Thunderbolt audio interface?

Or is this a Firewire audio interface connected via a (TB3-to-2 adapter) -> (TB-to-FW800) adapter chain? I read in other threads that Apple removed Core Audio Firewire support in Ventura. Without it, plug-and-play Firewire audio device that used to work without drivers will be left high and dry.

Firewire Audio for macOS Ventura – Apple Community

MacTracker says that the Mac Studio (2022) – the one based on the M1 Max or M1 Ultra – originally shipped with macOS Monterey 12.3.

If you clone a system from your M1 Mac mini, it needs to be at least that recent. The M1 Mac mini can run macOS Monterey 12.0 – 12.2.1, but those versions are not new enough for the Studio.

The latest version of Monterey is 12.7.2.

Since you have not explicitly mentioned the Adapter and its quirks, allow me:

Attempting to interconnect ThunderBolt-2 and Thunderbolt-3 devices requires the US$50

Apple Thunderbolt-3 <-> ThunderBolt-2 adapter

... plus likely a cable on one side or the other as needed. This adapter does not supply POWER across the divide, so it supports only self powered devices, and ONLY Thunderbolt, not any other kinds of devices, even with additional adapters.

Grant Bennet-Alder wrote:

Since you have not explicitly mentioned the Adapter and its quirks, allow me:

Attempting to interconnect ThunderBolt-2 and Thunderbolt-3 devices requires the US$50

Apple Thunderbolt-3 <-> ThunderBolt-2 adapter

... plus likely a cable on one side or the other as needed. This adapter does not supply POWER across the divide, so it supports only self powered devices, and ONLY Thunderbolt, not any other kinds of devices, even with additional adapters.

A question about power. The OP said that:

"I have just bought a second hand Mac Studio 2022 and cloned it successfully from my Mac mini M1. Everything seems to work fine except my Thunderbolt audio interface - there is simply no connection registered in the OS. Moving the cable back to my Max Mini and it works."

This suggests that the OP's Thunderbolt 1/2 audio device was self-powered, at least when connected to the M1 Mac mini.

Mac Studio with the Max processor has USB-C ONLY ports on the front, limited to 10 G Bits/sec.

Only the Back ports are ThunderBolt. Have you tested with the back ports?