From the worker perspective, he'd have to file his taxes in Portugal. He'd be a resident, because he is in the country for more than 9 months every year. Strangely, however, nothing prevents him being a worker for an entity outside the country (free movement of people, goods and services in the EU market mandate on barriers here).

You'd be in a strange split-brain situation, requiring legal advice. I imagine you could file income taxes in Denmark, and these would be deducted from the worker's Portuguese due taxes under the double taxation prevention agreements.