Recently the environment at my job has become rather toxic, with two members of upper management having an open feud with employees picking sides and engaging in shouting matches. I want absolutely no part of this, and, as such, I have begun looking for employment elsewhere. However, in the two interviews I've had this week, both interviewers asked (understandably) why I was leaving my current position after just 1 year. I've always been told that interviewers want to hear not why you want to leave your current position, but why you want to take the new position. Is this an exception? When explicitly asked why I'm leaving my current position, is it allowable to mention a toxic work environment, or should I stick with an easy excuse, such as "I am looking to increase my responsibilities"?
You should stay focused on telling the story of what you're running after, not what you're running from.
Badmouthing the current employer might not work in your favor - some might see it as a sign of what you might do to them in the future.
If the HR person probes deeper about your reason for leaving, you can keep it high level like "the culture of that organization was not a good fit". If asked even further, politely decline by saying you prefer not to divulge more details out of respect for the company's reputation. In my opinion, this would be appreciated as it shows maturity and your ability to remain discreet, and not get negative.
To answer your first question, no I don't think it's unusual at all for an employer to ask why you're leaving your current position (or why you left your previous position if currently unemployed) in an interview. My current employer asks this of everyone we interview if they aren't straight out of school.
Regarding the main question, I would actually somewhat disagree with the current answers. While I agree that you shouldn't delve into too many details and come off as badmouthing your current employer, I don't see anything wrong with mentioning that your reason is that the work environment has become bad due to two feuding members of upper management. That answers the question and lets the prospective employer know that you have a legitimate reason for leaving (i.e. you weren't fired for bad behavior or poor performance, etc.)
A candidate we interviewed recently had a situation very similar to what you described. He was working for small company where the CEO and the engineering manager were husband and wife... and then went through a messy divorce. When asked this question, the candidate told us exactly that and that the CEO was now intentionally causing problems with engineering just to spite the engineering manager. That's a good reason for an engineer to want to leave a company, IMO. The candidate didn't go into any details of exactly what all had been going on or badmouth either the CEO or the engineering manager; he just gave us the above information, which sufficiently answers the question. None of us found his answer to be at all unprofessional and we certainly didn't hold it against him in any way. Our CEO just assured him that he isn't married to our engineering manager, so he wouldn't have to worry about that situation arising at our company. - lol