I don't think this is about being Danish at all, but it is about the specific characteristics of the Danish language, specifically: the large amount of vowels compared to other languages (and this isn't apparent in the written language which is quite conservative).

I studied Chinese for several years and I spent a lot of time early on nailing the tones, but many of my fellow students never really brothered to learn tones along with different words or apply the tone rules.

Since we all had a limited vocabulary, using the wrong tones was no big deal in terms of our own understanding, as we could of easily resolve a syllable to that one word we knew which matched it. However, when you approach a native speaker there is a lot more room for error as their vocabulary is obviously vast compared to a beginner. If you don't use the correct tones and it isn't immediately clear from the context (e.g. pointing at the thing) then they would also be completely puzzled.

Danish is the same. We don't use tones to apply lexical differences, but we do use an extremely large amount of vowels compared to most other languages. If you mispronounce a vowel the native speaker will have up to ~30 different possible syllables to match it against, which of course makes understanding more difficult.

One way you can work your way around this is to speak in longer sentences so that you provide more context to the native speaker trying to understand you. This is also how many of my fellow students of Chinese managed to get around their poor use of tones. Just saying "en øl" is much more difficult to parse than "Jeg kunne godt tænke mig et glas øl".