Are "stout" and "mild" the same thing - "mild" being popular in the UK in the post-war decades.
The short answer is no. While they're both styles of dark beer nowadays Mild is very low ABV with a lot of malty/nutty flavors and fruity sweetness. Stouts are stronger than milds with no fruitiness and a lot of roasted and burnt flavors instead of malts. Most will probably be more bitter and have less residual sweetness than milds.
The long answer is:
Historically stout and mild were completely unrelated terms that referred to completely different things than they wound up being known for. Both started being used in the 18th century. Stout originally just meant strong and Mild was used as the opposite of Keeping (or Stale), which basically just meant the beer was fresh rather than being aged in a barrel for any length of time.
Mild turned into a strength descriptor in the early 19th century when brewers started using the X, XX, XXX system to classify their ales, which referred to both the ABV and pounds of hops per barrel. But Mild was still a pale beer. As the 20th century progressed Milds started becoming more amber colored and playing more as malty and sweet.
All beers until this point were actually much stronger than we're used to today. Barclay Perkins X Ale in 1839 was actually close to 8% ABV.
Grain shortages and government restrictions towards the end of WWI dropped Mild to close to 2%, and brewers also added dark sugars and mixed in amber and brown malt to keep some measure of flavor. After the war ended ABVs bounced back up but new taxes in the 30s pushed them back down, close to the modern range for Mild. Dark mild being the standard and Pale mild being a variant didn't really happen until the 1950s.
The history of stout is quite a bit muddier, though the general consensus seems to be that it evolved out of Stout Brown Porter and has been largely interchangeable with Porter for most of its history.
The differentiation probably started with in the invention of Black Patent Malt in 1817. In the late 1700s new technology had been invented that allowed brewers to track their efficiency and they realized the roasty brown malt they used for porters was really expensive compared to how much fermentable sugar it produced. So they started making porters out of mostly pale malt with brown malt and burnt sugars among other ingredients for color and flavor.
Britain had banned the use of anything besides malted barley in brewing in 1816 which really kindof screwed porter brewers over. With Black Patent they had something which was absurdly dark enough that they could finally make up the color difference using pale malt. Though most brewers in England continued to use some portion of brown malt, brewers in Ireland switched completely to pale and black malts only. Irish beer was also less hoppy than English beer in general since hops didn't grow as well in Ireland.
Guinness renamed its XX Porter to Extra Stout Porter in ~1820 and the popularity and marketing gradually dropped "Porter" from the name, which is probably where Stout began to evolve as a style separate from Porter. But it's in a bit of grey area, people in 2015 are still debating Porter vs Stout.
Short version of the long story
Mild originally just meant Fresh and as a style evolved from the weakest pale beer a brewery made, and only really became what we think of it as today in the last 60 years. Stout originally just meant Strong and was a synonym for Porter for a very long time until marketing tried to differentiate it maybe 150 years ago.
Sorry about the wall, but I fell down a rabbit hole of interesting beer history.