If I say I'm Dutch, that does not mean that my people were from Denmark. It means they were from the Netherlands. The Danish* are from Denmark.
To further clarify: Holland, though synonymous with the Netherlands is actually just a region in the west of The Netherlands, whereas the Netherlands** refers to the entire country.
There. I'm glad we got that out of the way.
*They are also delicious breakfast treats.
**If you want to be superproper, it is The Kingdom of the Netherlands.