Throughout alot of Halo, many bring up the power of the UNSC. A good example is in Nightfall, Aikien says that the UNSC wishes to expand its power. But the UNSC is just the military branch of the UEG. You can argue that the UNSC has alot more responsibilities as Frankie seemed to indicate that the CAA no longer really exists. Why do so many people put the blame on the UNSC when it’s merely an extension of the UEG, you know, the government telling the UNSC what to do?
In many cases it’s because characters understand that real power lies with the UNSC, not the civilian government. This was the de facto state of affairs throughout the Insurrection and during the first years of the Covenant War. The UNSC then instituted martial law that didn’t end (nominally, at least) until 2553. It’s understandable that those who have a bone to pick with the powers that be would blame what they view as a domineering (perhaps even fascist) superpower-that-officially-isn’t. Besides, plenty of them blame the military specifically (not as an erroneous generalization) for one reason or another, often in the “They left my planet to die” sense.
On a related note, there actually has been a notable increase in references to the UEG over the last few years. Kilo-Five is a good example, as is Escalation. Even UNSC Navy service and dress uniforms are now emblazoned with the UEG globe and laurels.
The Universe’s section also points out the fact that the UEG is controlled by shadow governments (ONI), so it is still an empty shell in many ways. And as said before, repression mainly comes from the military, hence the blame on the UNSC.
I don’t think the UEG actually tells the UNSC what to do.
I was going to be explaining how the UNSC took over the Government due to the Insurrectionists and then the Covenant, but everyone above pretty much explained it before I could. So all I can really say is that the people above hit it dead on.
Basic the Kilo-Five Trilogy states that cause of the Human-Covenant war. The UNSC basically became the de-facto leader of the UEG, even though the UEG still has a president as seen in halo Escalation. But the USNC, mainly ONI, are callIng the shots and the Innies know this. Why they call the UNSC fascist.
So many good explanations I was too late like the others said 
The UNSC issued martial law until 2553. I’d imagine most people are used to recognizing the UNSC as the head still. Also the UEG has no real power without the UNSC, so there’s that.