Fans chanted "Manolito, Manolito" like Real supporters did when the 35-year-old Togo international played in Madrid.
Adebayor, who also played for Arsenal, Tottenham and Crystal Palace during a decade-long stay in the English Premier League, held up a banner in Olimpia's colors saying "I'm already here," as fans chanted "this is your public".
"I'm part of the Olimpia family now. I have to follow their dream, I have to follow their target, which is to win" the league and Copa Libertadores, Adebayor said in a video message to fans.
Adebayor will be available for Olimpia's 2020 Copa Libertadores campaign in which they face Brazilians Santos, Defensa y Justicia of Argentina and Ecuador's Delfin in Group G.
Olimpia are three-time former winners of the Libertadores and have won the last four Paraguayan titles in a row.
Adebayor is linking up at Olimpia with former City team-mate Roque Santa Cruz, 38, who was top scorer in the Paraguayan league last year.
"Adebayor hasn't come here on holiday," said Olimpia president Marcos Trovato.
Olimpia treasurer Miguel Brunote said the Togo forward was commanding the biggest ever salary in Paraguayan football.
Brunote said Olimpia beat off competition from Argentine giants Buenos Aires to sign Adebayor, who had been a free agent since December when he left Turks Kayserispor.