Simply mount your time machine backups volume then navigate to /Backups.backupdb//Latest. Under there is your latest backup. You can then copy and paste the files manually wherever you want them.

Just Checking that you have not missed the "holding option key down in time machine" thing that I missed for ages:

Source: http://www.macworld.co.uk/feature/mac-software/complete-guide-time-machine-mac-backup-3626572/

Complete guide to Time Machine: Can I browse Time Machine backups from another Mac, or older Time Machine backups from my own Mac that I no longer use? Yes. Attach the disk, or attach the Time Capsule to your main Wi-Fi/internet router, then hold down Alt (Option on some keyboards) on your Mac and click the Time Machine menu bar icon. Select Browse Other Backup Disks. The Time Machine disk should appear in a list, and once you've selected it you'll see the standard Finder-based Time Machine view where you can move back in time to retrieve older files.

If the backup is encrypted you'll be prompted to enter the password before you can gain access.

This worked well for me so thought I would share here.

The backup they gave you; is it just files or is it the Backups.backupdb folder?

The Time Machine backup you have is "tied" to the old computer. You do have a couple options to restore the data however:

Link Time Machine To Your New Computer

Check out this other question on SuperUser: 4906 for how to migrate your Time Machine backup to work on your new computer as well.

Browse Other Time Machine Disks

Hold the option key and click on the Time Machine Menu Extra and choose "Browse Other Time Machine Disks".

Update: I have modified the first paragraph and added a section at the end.

This answer was accepted whilst limited to restoring your data, as in the question title. It now also covers associating the restored disk with the existing backup so that future backups continue without duplicating disk space on the Time Machine (TM) disk.

Firstly, forget everything you have read about Backups.backup which only applies to backups where the TM disk is using the HFS+ format.

From Big Sur onwards the format for new backups is APFS. APFS formatted backup disks use snapshots for each backup. Each snapshot is an APFS volume - it is not a directory.

An aside: The man page for tmutil covers backups in both formats with no clear distinction between them. As a consequence, it can be very confusing except as a command reminder for those who already understand TM in some detail.

Before doing a restore, turn TM backup off. Just so that this doesn't confuse or impact performance while you are restoring.

Use Finder to navigate the TM disk. In Finder your Media TM disk will look something like my MyBook6B-TM:

Each backup is a snapshot with a “disk & clock” TM icon.

Choose the one you want to recover from and navigate it in the usual way with Finder. Then just drag and drop folders or files from there to another Finder window with your new disk.

Drilling down can be a little slow at times with a beachball - be patient.

At this point you should on your way to recovering files from your Media TM backup disk.

As a little more background, behind the scene, the snapshot has been mounted (like it was a disk). In the Finder window above I could hover over the /Volumes/.time... at the top to see the mount point where the snapshot has been mounted. It is this location that you can (if needed) use in Terminal by typing cd and dragging the disk-like icon to Terminal. This is what I have done below. I did not type the /Volumes....

gilby@Beth/Users/gilby% cd /Volumes/.timemachine/02EED3F7-B8DF-44BB-8EA6-28D7822B1A4B/2022-11-15-215052.backup/2022-11-15-215052.backup
gilby@Beth/Volumes/.timemachine/02EED3F7-B8DF-44BB-8EA6-28D7822B1A4B/2022-11-15-215052.backup/2022-11-15-215052.backup% ls -lah
total 136
drwxr-xr-x@  5 root  wheel   160B 15 Nov 21:43 .
drwxrwxr-x@  5 root  wheel   160B 15 Nov 21:50 ..
-rw-------   1 root  wheel   806B 15 Nov 21:50 .com.apple.timemachine.checkpoint
-rw-r--r--   1 root  wheel    62K 15 Nov 21:43 .exclusions.plist
drwxr-xr-x@ 20 root  wheel   640B 15 Nov 21:50 BethSSD - Data
gilby@Beth/Volumes/.timemachine/02EED3F7-B8DF-44BB-8EA6-28D7822B1A4B/2022-11-15-215052.backup/2022-11-15-215052.backup%

This only works correctly because the snapshot has been mounted due to Finder asking to see what is inside the snapshot. It can be mounted in other ways, but via Finder is easy.

A final point coming back to what you have already discovered. The /Volumes/.timemachine folder contains the mount points at which backup snapshots can be mounted. But navigating with cd and ls is not sufficient to mount the snapshot.

Associating the new disk with the existing TM backup:

I was hesitant to describe how to associate the new disk with the TM backup so that future backups would just add to the existing backup without duplicating everything. But thanks to @Christoph being prepared to take risk, we have established how to associate the disk. This addendum is taken from the discussion in comments.

The man page for tmutil does give the commands for associating the disk, but the example is only correct for HFS+ format TM disk. There is no example for APFS.

In this case the destination TM disk is formatted APFS and correct command (specific to the configuration in the question) is:

sudo tmutil associatedisk "/Volumes/All Data" "/Volumes/Media TM/2023-01-13-202029.previous/All Data"

For others wanting to do this replace the All Data with the name of the recovered data disk.

The second parameter is not so easy, but see the last two commands in the question which show the results of ls -la "/Volumes/<TM disk>" and ls -la "/Volumes/<TM disk>/<date-time>.previous".