"From today, I am Polish," Santos declared at the official ceremony in Warsaw.
"It is an honour to continue my work as a trainer here," he added.
Santos, 68, succeeds Czeslaw Michniewicz, who was sacked last month following a last-16 appearance at the World Cup in Qatar.
Poland finished second in their group, beating Saudi Arabia, drawing with Mexico and losing to eventual champions Argentina.
In the last 16 for the first time in 36 years, they went out after a tame 3-1 defeat at the hands of eventual runners-up France.
Lisbon-born Santos quit as Portugal coach in December, following the team's shock World Cup quarter-final loss to Morocco.
Appointed coach of Portugal in September 2014, Santos won the country's first major title at Euro 2016 and followed it up with the 2019 Nations League.
However, his time in charge also saw last-16 exits at the 2018 World Cup and Euro 2020.
"The choice was difficult but we have chosen the best," said Cezary Kulesza, head of the Polish football federation.
"Our first objective is to qualify for the European Championship" which will take place in Germany in 2024, he added.
Santos also coached the Greece national team from 2010-2014.