I don't see how the second version (including a subdirectory of the language in both cases) would have any negative (or positive) SEO impact over the first version (without a subdirectory for the primary language).
The important criteria is that the different language versions have different URLs. Both your URL versions satisfy that criteria.
The subdirectory name is just to make the URL unique. For direct-SEO it doesn't even need to be representative of the language - the HREFLANG attributes handle that. Instead of
/da, you could use
/foo and I don't think the search engines would really care. However,
/foo isn't so meaningful for users, so using something that is representative of the language (as you are doing) is obviously better for users (which could indirectly benefit users).
So, I don't think there is any difference between these two URL structures SEO-wise.
the domain is Danish
.dk, which makes it implicit that it is Danish
Well, yes, although strictly speaking the domain is Denmark (locale / geo-target).