While there is no check, there is information. Nowadays this is all electronic, but there were other means before that.
For instance, in the US, visitors used to have a part of their I-94 stapled into their passport when they entered the country. On exit, the check-in agent would take it from the passport, and all collected I-94s would be send to the relevant agency. Of course it was extremely inefficient (I have no idea what actual processing was done with those forms, honestly).
Nowadays this is mostly all electronic. Airlines provide manifests (lists of passengers) with more or less information to the relevant authorities, which try to match them to the relevant record (which is straightforward in most cases even with very minimal info, but can be more difficult in special cases like people having multiple passports, possibly with different names, or extremely common names...).
Visitors to the US can see the result by checking their own I-94 online after leaving the US: the exit will (should?) have been recorded even though they never met a CBP agent on their way out.
For a long time there were a number of "holes" in the system, for instance:
In the UK only airlines provided the info (and I think not even all of them). Train (Eurostar), ferry and bus companies didn't. This has been addressed since.
There was a hole for people exiting the UK via Ireland. Since the UK and Ireland are part of the CTA (Common Travel Area), exits to Ireland were not recorded, and either Ireland did not record exists or did not report them to the UK, don't remember which. I think this has been addressed as well.
Also, note that some of the flights in your friend's itinerary (FR-DE and IE-UK) are internal flights/trips (Schengen for the former, CTA for the latter), and don't necessarily have border control at either end anyway.
However there IS exit passport control out of Schengen, so your friend did nearly certainly meet a border agent when they flew from Germany to Ireland. They should have a matching stamp in their passport. If they don't, they should keep evidence that they actually left the Schengen area on that date (boarding passes, hotel receipts in other countries), because those stamps are currently the only record of Schengen entries and exits.
US-France-Germany-Ireland-London-US:
- US exit: no check
- FR entry: check
- FR-DE: internal, no checks other than ID
- DE exit: check
- IE entry: check
- IE-UK: internal, no checks other than ID
- UK exit: no check