Aerith remembers him, and so does Tifa. Cloud re-remembers him after he's recovered, he's not really pretending to be somebody else, more like the Jenova cells and Mako poisoning with a dose of trauma have messed up his mind and brain and he has Tifa and Zack's memories confused with his, and Cloud acts how he thinks a professional Shrina Inc Solider would act like, like Sephiroth, cold and heartless. Because he has Zack's memories mixed up with his, he thinks that he was the Solider that went to Nibelhiem and was the Solider that was 1st Class, like Zack was.
It's more like that he truly thinks that he's somebody else, until Sephiroth and Hojo gaslight him, into believing that he's a nothing but a Sephiroth clone and leading him into a mental breakdown. Then when Tifa restores him, Cloud remembers his true identity and his friendship with Zack Fair.
Cloud fully believes that he's an ex-professional top class Solider during the first part of FFVII, no pretending. Also, Tifa and Aerith are too scared to tell him the truth about it. If you take Aerith and Tifa back to Gongaga, they will both mention that they know about Zack.
I forgot about Tifa, she did just go with the flow when Cloud started spouting nonsense. But in the grand scheme of things, especially looking at Crisis Core, they implied Zack should have had a much bigger role than he ended up having. I'm really hoping some of that changes in the FF 7 remake. Zack is my all time favorite Final Fantasy character and I'm extremely excited to have him back in any format. I would have preferred Zack stayed alive myself.