In Inception, Robert Fischer was kidnapped by Cobb's team who had him in Taxi with Eames, Arthur and Saito. Robert must have known that they are kidnappers as they kidnapped him anyway. But, in later stages and further dream levels he accepts them as allies without any resistance. He was told by Mr. Charles that they are his allies (Eames, Arthur and Saito). But, was that just enough to be convinced? Why did he never doubt Cobb as his enemy/kidnapper/extracter? Was Fischer so scared that he couldn't figure out that the people who kidnapped him became his allies and no suspicion arose?
In the Real World, Fischer wasn't kidnapped, but sedated on an airplane. As far as he knows, he drifted off to sleep naturally, with nothing to fear. The kidnapping occurred in a dream, and like any dream you or I might have, there's a tendency to just "go with it" while you're dreaming. In reality, you would NEVER trust a skeezy fat guy in a white panel van offering you candy if you get inside. In a dream, however, you might just go along with it (and there might even be candy!)
After that, each new layer of dreaming is like a dream to the dream-person in the previous layer. A dream within a dream, quite literally. The only people in full cognitive control are the ones hooked up to the machines; aka Cobb and his gang, in this case.
As for the projections; Fischer had "training" to "resist dream infiltration" but he likely had no idea what that resistance would look like in an actual dream. It's not like somebody told Fischer in reality "look out for the guards dressed liked this and obey them no matter what because they're your protector projections." For one thing, a dream can take ANY form, and the projections that resist infiltration would naturally take whatever form is most appropriate for the dream, so they aren't static. For another thing, if the projections were static or had a static symbol, an infiltrator could theoretically learn what that symbol is and mimic it after he gets inside your dream. That would be a weakness in the training program. The projections operate at a subconscious level to stop the infiltrators no matter what the actual dreamer wants or does, this is the only way to make sure they can't be corrupted themselves.
So, in short, since Cobb and his team kept Fischer close on each level of the dream, they were able to manipulate "what happened" in the dream and tell Fischer what they wanted him to do or think about. Fischer "went with it" because he was dreaming without a dream machine, and as long as Cobb could keep Fischer away from the projections, they couldn't stop him. This is ultimately what inception is all about, after all. Making someone think an idea was theirs when it really wasn't.
If you prefer to take the position that the entire movie was a dream - including the layer that Cobb thought of as "reality" - then Fischer himself becomes just another element of Cobb's dream. Fischer does what's necessary to keep the dream going as a figment of Cobb's imagination, and Cobb "just goes with it."