Huh, that really does seem like it'd bias the jury -- you've got him dressed like a convict even though he hasn't been convicted of anything. The guy might as well be in a skull shirt. It's crazy that they do it this way!
Which is precisely why in real life, they don't. The Supreme Court has ruled that the State isn't allowed to make a defendant wear that in court. A prisoner may choose to appear that way, if for some reason they want to bias the jury or just don't know what the hell they're doing. But preventing a defendant from appearing in the dehumanizing garb of a prisoner is so crucial that public defenders often hold clothing drives to make sure their clients can dress up. This is Defense Attorney 101 shit here, but both Nelson and Murdock appear to have missed that class.
Though it is tough to find suit jackets with a skull on them.
There's also a thing really common in TV court dramas, which is that cases like this seem to go to trial before the victims have even finished dying. The judge gives Nelson and "Why does my suit have eye holes" Murdoch one week of pre-trial preparation time. High-profile trials (like ones for guys who have shot lots of people) are obscenely slow precisely because they require an absurd amount of prep work. For example, Aurora Theater shooter James Holmes was arrested in July 20, 2012 and didn't even start discussing plea deals with the DA until March 27, 2013 -- more than eight months later. His actual trial didn't start until two freaking years later. Just the process of selecting the jury took three months!
Once the trial starts, Nelson and Murdoch are railroaded by a clearly biased judge. There's no doubt that sort of thing happens in real life, but the system does account for the fact that judges are human and might be drunk that day. For example, she unfairly strikes the defense team's key testimony (from a medical examiner who admits falsifying records). Here's where a real lawyer objects -- not to get the judge to change her mind, but to preserve the issue for appeal (such an error on the part of a judge is one of the few things that can get a verdict overturned).
The legal system does love its do-overs.
Instead, Foggy and Matt sit on their thumbs, essentially erasing all possibility of ever using the cornerstone of their defense again. Later, a guy in the gallery starts shouting that Castle killed his father, which hands them a chance to request a mistrial. Two guesses as to whether they bother.