The first thing to remember is that form follows function. The guns and rifles of Great Britain were developed to support the functional needs of the British sportsman - both in the UK and in the colonies - particularly India and East Africa. The same was true of Germany and Austria (also France and Belgium - both of which produced excellent guns and rifles). It is important to remember that the German/Austrian colonial market collapsed with the end of WWI in 1918. That had a major effect on Continental development of dangerous game rifles.
In the UK, shotgun hunting was largely defined by the driven hunt - primarily grouse. The English game gun was created to accommodate that specific set of challenges; fast inbound birds, presenting targets near and far. It is why the traditional English SxS is choked very open on the right and fairly tight on the left in order to provide that instant choice. Guns for "rough hunting" - for walking up game or shooting decoying pigeons, etc - typically mirrored the game gun, but were usually of simpler and less expensive design for the farmers and small landowners who purchased and used them.
The rifle for hunting in the UK was largely defined by deer stalking. There are many exceptions, but that solution was typically found in either a falling block single (the Gibbs or Alex Henry are quintessential) or a medium bore bolt action. Those bolt actions and barreled actions were usually sourced from mauser. But one also finds Mannlicher Schoenauers and repurposed Lee and P-14 Enfields built as medium bore stalking rifles and carrying prestigious English names. The .275 (7x57) was the classic choice, though the .308 has gained a great modern following.
Again, there are exceptions. For instance, there were a whole class of rifles used for pests that are called "rook" rifles. These were built on falling block, Martini, and break-open designs and were chambered in what we would consider "pistol" cartridges.
Georgian taste was restrained, and this notion of minimalism but unsurpassed quality manifested itself in British stock design - essentially nothing present that didn't contribute to accurate shot placement.
Like the British gunmaker, the Continental gunmakers delivered designs to meet the needs of their customers. The German/Austrian hunter also participated in drive hunts. However these were often lines of beaters and guns marching across fields. Other times game was pushed toward them, but the bag was typically mixed. A hunter could be presented with a flushing pheasant, Hungarian partridge, a European hare (weighing as much 8-10 lbs), or a wild boar, and all within the span of a few minutes. Guns for such hunting tended to be robust, tightly choked, and with dimensions more like a rifle than an English SxS. Combination guns were extremely popular and useful. Unlike his deer stalking colleague in the UK, the German or Austrian spent far more time in high-seats. On the typical revier (lease), the German hunter would have a detailed shooting plan requiring the selective take of age groups and sexes of deer. He thus spent days in blinds (being certain a particular doe is barren takes skill and effort), and was also constantly on the lookout for foxes, feral cats, martin, etc. for which a load of shot was far better choice. Combination guns reigned supreme. "Bockbuchsflinte" (shotgun over a rifle) or drillings (SxS shotgun over rifle) were the most common. They allowed the instant shot selection of whatever was most appropriate for the animal that appeared. Drillings were also extremely useful on drive hunts where wild boar might be encountered.
The use of a wide range of shot sizes led to the development of very strong actions with numerous locking mechanisms. Critics often denigrate these wonderful guns as over engineered, but they are robust and perfect for their intended use. The trigger plate action became as common in central Europe as the Anson & Deeley did in the UK. The Kirsten locking system was ideal for OU's and very strong. The transition from OU shotgun or Bockbuchsflinte to OU double rifle was easy and Austria and Germany built many wonderful doubles in this form (far more than the British did SxS double rifles). It was also a very easy step to drop everything but the rifle barrel creating the beautiful single shots we find today.
Hunting rifles were also medium caliber for essentially the same game as the UK. Stylistic differences were mainly found in the stocks. An exception were the petit Mannlicher Shoenauers that were created for Alpine hunting and the large number of 9.3x74R OU rifles which were so effective on boar and red stag in the thick cover of Central Europe (compared to the needs of the British stalker in the Scottish highlands).
Because Germany lost its German Southwest (Namibia) and German East (Tanzania) African colonies at the end of the First World War, much dangerous game rifle development ended. When Americans first appeared in Africa in numbers between the wars, it was the British interpretation of a dangerous game rifle that awaited them in the hands of their stalwart white hunter. It was either a heavy double or a robust mauser-based bolt action. Had Germany retained its colonies, I have no doubt we would consider a heavy OU double rifle every bit as "proper" as the English interpretation of dangerous game rifle, and our list of regularly used dangerous game cartridges would include a number of now nearly extinct metrics.
This is barely a Cliffsnotes "history." but the divergent but still parallel development makes sense when we remember form always follows function.
