Dogan, speaking to BBC Sport Africa and BBC Turkish, has openly admitted the numbers involved: Salah will earn €17m a year, plus 20% of any merchandise sold bearing his name.
Season ticket sales had been targeting 25,000 before the move, but the club now expects to pass 30,000, with major companies in Turkey, Europe and the Arab world reportedly approaching Trabzonspor about getting involved.
"Purely from a financial perspective, this transfer has no direct return as such," Dogan said. "The attention it has generated and the level of engagement it has created around the world cannot easily be given a direct financial value."
That attention was on full display when 35,000 fans packed Papara Park for Salah's contract-signing ceremony this month, a reception Dogan says convinced him the club had pulled off a coup.
"He's the biggest Muslim footballer in the world. He's an influential role model," Dogan said.
The move raised eyebrows given Salah, who sits fourth on the Premier League's all-time scoring list, had been widely tipped for the Saudi Pro League or Major League Soccer after nine seasons at Liverpool.
Dogan says Trabzonspor had been pursuing him since the World Cup, waiting their turn under what he described as an unwritten rule among Turkish clubs not to compete for the same target simultaneously. Once rival domestic interest fell away, talks moved quickly: an initial Zoom call, a pause while Salah holidayed in Mykonos, then further calls before a deal was struck within five or six days.
Trabzonspor, based in the Black Sea city that shares the club's name, have won the Turkish title only once since the early 1980s, and sit outside Champions League contention this season after a first-leg Europa League play-off defeat to Ferencvaros. But Dogan, who took charge in 2023, argues the city itself is the pitch.
"We're the biggest one-club city in the world," he said, describing how he sold Salah on Trabzon rather than trophies.
The club brought Salah to Istanbul rather than Trabzon for his medical, a deliberate choice, Dogan says, to show him the reception he would get across the whole country rather than just one city.
Thousands of fans turned out at the airport in the middle of a working day, and Salah made his debut there too, coming on as a substitute in a 1-1 draw with Kasimpasa on the opening weekend of the season. He is expected to make his home bow in Trabzon on Sunday.
The signing forms part of a wider strategy at the club, which sold Ivorian midfielder Christ Inao Oulai and Brazilian forward Felipe Augusto for large profits this summer after just a single season each, reinvesting in young talent from Albania, Belgium, Cape Verde and Portugal alongside a returning Andre Onana, back on loan from Manchester United.
"We're working to make Trabzonspor and Trabzon itself a destination for talented young players from all over the world," Dogan said. "Mohamed Salah is a really great example in this respect."