Cloud's outfit it canonically black. It's been shown that way in the Compilation of FFVII and stated as such by the creators.
That being said, I don't think its that weird given the circumstance of Cloud's adventure of FFVII, the areas he frequented, and the climate of the area.
For one, thanks to the hell Genesis, his copies, and SOLDIER raised, not many people saw SOLDIER in a favorable light anymore. Many of the fanclubs disbanded and were probably forgotten. Not only that, but as shown in DC, Shinra went through a very sweeping and viscious silencing campaign to erase all records of "G" and his legacy out of the public. As the G Reports stated, only a select few of older, slum dwellers still would whisper his name and ever mention him. He in the end, became something akin to an urban legend.
Given the fact of the adverse reaction and information block out and censorship of SOLDIER, most people probably forgot about it and purposefully put SOLDIERs like Zack and the others out of mind. It's not like many people personally knew Zack and if they did, they wouldn't dare say anything.
The only people who would've definitely raised an eyebrow would be the Turks. Specifically, Reno, Rude, and Tseng. The fact they were close to Zack, and would see Zack's iconic sword and outfit, being worn by someone else would definitely cause them to react. Of course, this isn't seen in FFVII at all.