That's why, when you're interviewing candidates, your first job is to get them to relax. Start out with small talk and softball questions until you sense them unwind. Then move slowly into the real questions.
When you get to the coding problems, again, start with dead-simple questions so that the candidates have the time to gain some confidence before the harder problems arrive.
The first programming problem I give – and this is after some small talk – is usually something like, "Print the odd integers between 1 and 100, in increasing order." This problem is so simple that it doesn't tell me anything about the candidate's ability, but that's okay. It helps to calm the candidates so that when I get to the problems I care about, I'll get better measurements.
That's the goal: To measure how the candidates actually perform, not to see whether they'll choke under interview stress.
Not really. A job interview is your one single chance to make a good enough impression to get the job. You could freeze up in a meeting and it would be fine, if the rest of your performance is on par.
One meeting in a job is a very tiny part of your overall employment. An interview is the one and only impression you get to make.
(For further reading, please consult the lyrics to Eminem's "Lose Yourself". Mom's spaghetti.)
How exactly? Freezing up in a meeting can get you fired just like freezing up in an interview can get you turned down for the job. Sure, you'll get comfortable with your coworkers, and they'll get more forgiving of temporarily lapses, but I think most interviewers (for non-terrible jobs especially) purposefully give some extra leeway to interviewees, since it's understandable for them to be a bit more nervous than a coworker would be.