Possibly. So far (season 2 completed) there is no evidence to the contrary.
Warning: major spoilers up to and including season 2.
The details that we see of the alternate universe correspond to our history. The Cuban Missile Crisis (and its resolution) are shown, the book Lolita, a history book of World War II as we know it.
In episode 10 (season 2's finale) we find out that Tagomi's assistent Kotomichi is from the alternate universe: he was in Nagasaki when it was destroyed with a nuclear bomb, after Hiroshima. (No date is given for either bombing.)
[...] but then they show that the Japanese were able to develop a hydrogen bomb and test it in their territory.
This is incorrect. In the alternate universe, at Tagomi's home, an 8mm film is played showing the detonation of a hydrogen bomb over the Bikini atoll. Tagomi's son says "we" tested the bomb but he considers himself American, not Japanese. He also says it was done to show "the Russians" that they had the bomb. In the MitHC universe the Russians were defeated by the Japanese and Nazi Germany and are not a world power.
Later, Inspector Kido says that this bomb was detonated in "our" territory, meaning Japanese territory, but that is from his perspective (episode 10: Fallout). In the MitHC universe the Bikini atoll is Japanese territory but in the alternate universe, where the bomb was actually developed and detonated, it obviously isn't.
Later on, Smith takes the film to Berlin in a last-ditch effort to stop World War III. Smith knows it's a film from the Man in the High Castle but he hides this fact to make the Nazis think the Japanese have the hydrogen bomb. Even though a Nazi himself, he does this to avoid the war because it would be fought mainly on American land with massive American casualties.
Although it is possible that the alternate universe as depicted is not our universe I have not seen any evidence to support it.
Aside: Despite the incredible attention to detail regarding the production design, there was one major error made in the pilot. An unforgivable error, in my opinion.
Juliana is shown to be an aikidoka and after her test/demonstration a Japanese fellow student talks to her. He is wearing his dogi (training jacket) incorrectly: right-over-left (like a women's blouse). No budoka would ever wear his jacket like that, let alone a Japanese budoka. The only time a gi is worn right-over-left is during your own funeral.
Upon rewatching the pilot I noticed another Japanese aikidoka wearing his dogi like that. Both aikidoka wear hakama and are surrounded by other hakama-wearing aikidoka so there can be no excuse for this mistake.
And then there is the pronunciation of yakuza. Both Americans and Japanese pronounce it as yaKOOza instead of YAkuza.
I have not noticed any other glaring errors but if these are not errors but intentional discrepancies (which I highly doubt) then it could be that the Nazi/Japanese-dominated world is an illusion or hallucination experienced by Tagomi.