A further thought: Since we did not find Harry in the Danish census 1880, in any area of Denmark, nor find him born in any parish in the city of Aalborg, or in any parish in Aalborg amt as a whole, maybe you are having the same lack of luck.
The next step will be for you to concentrate on USA documents. You will want to obtain his marriage certificate, World War I draft registration, naturalization papers, and death certificate, if you do not already have these.
If you want further help from the forum, you can share:
1) what you find in the above documents
2) what is your source for his birth date
3) what is your source for his place of birth.
At this point it seems likely he was not born in Aalborg, though he may have resided there at some time. The 1910 census gives his birthplace as Germany. Many times when immigrants identify as Danish but state their birthplace as Germany, they were born in the Schleswig-Holstein (Slesvig-Holsten) area of southern Jutland - those lands between Germany and Denmark.
Note that Harry immigrated in 1887 at the very young age of 14 (if his birth date and the US census records are correct). His first work in the USA census is "mariner sailor." Thus it could be that he came to the USA on a ship doing a job usually reserved for a young man - such as cabin boy or kitchen crewman. It is possible he left his employment or abandoned ship once in America. This could explain why we also have not found an immigration record for Harry. Just a theory.
Alternately, we can guess a boy so young would have traveled with family or traveled to a family member already in the USA. Does your family history lore include relatives of Harry's in the USA? An oral family history is a sturdy source, and I consider it significant that you have mentioned no parents or siblings in America.
Searching with wildcards for many variations of his name has not brought results. Of course if Harry's name is really very different in Denmark than in the USA, that too could account for his AWOL status.
All of which points to the importance of developing any and all possible documents about him in the USA.
Hope these comments help - they grow out of a collaboration with another volunteer researcher.
Here's a DIY kit for Danish genealogy, in a thread in this forum,