IANAL but as far as I know in the EU people have to pay their taxes based on the country where they live (residency), not based on nationality. Since you spend more than 6 months a year in the US, you're considered a US resident so you're no liable to general income taxes in Portugal.
Per the official web portal of the EU:
Living in Portugal? You must pay tax on your worldwide income there.
Living in Portugal for less than 183 days? You are not considered tax
resident and pay tax only on the income earned in Portugal.
You're considered tax resident in Portugal if: (1) you lived there for at least 183 days (consecutive or not) during a
tax year or (2) you lived there for less than 183 days but on 31 December
of the year in question, you had a house/flat in a condition that
suggests you intend to maintain and occupy it as your usual residence.
There might be other local taxes such as property tax (if you own a property in Portugal) and/or residence tax. And of course you would pay the VAT on things you buy in Portugal, but that's transparent.