There is no "special box check-in" process. There are however various procedures for non-standard or outsize baggage that vary from airport to airport.
Checking in a box of "presents" that you didn't pack yourself, by way of request from a third party, that were "fedexed" to your hotel, is clearly likely to arouse interest.
What were you thinking? Since they were being "fedexed" in the first place, why didn't you simply pay the required amount, and have them freighted directly to you? If they were accepted at check-in after primary inspection, your friend presumably answered "yes" to the question "did you pack this luggage yourself?"
If anybody carries something in their luggage for somebody else, they do so on the basis that the items are effectively theirs for the purpose of carriage, and they accept full responsibility for those items as regards the security and inspection of those items, as well as any customs, duty or other considerations that may apply.
A cardboard box at check-in doesn't in itself usually involve any additional screening procedures unless it is particulary fragile, heavy, unusual size or shape, or require special handling or labeling.
Similar procedures are in place in many other countries as well as the UK.
It is not particularly clear from your post if your friend inspected and accepted the contents of this "package" prior to checking them in. However, it is comforting to know that a cardboard box delivered to a hotel address in London and being carried on board an airliner on behalf of somebody else, is (and the courier) pulled for additional inspection.
Based purely on what you have said, it is good that secondary security stopped the package and passenger from boarding until they were satisfied that no risk existed. It is unfortunate that your friend was inconvenienced in this manner, but he has probably learned a lesson. It is unfortunate that you didn't consider the implications that can be inherent with saving a few dollars on shipping costs.