I'm a hiring manager, and I know my HR recruiters regularly ask this question (I know, because they tell me the responses).
I know that YMMV depending on where you're interviewing, but I will tell you that I do use this information to frame my offer and the priority of the offer paperwork with HR. They are a resource constrained organization like most, and so being able to tell them, "Hey, we need this in two days" only when you really do need it that quickly helps a lot.
It will also affect how the HR recruiter talks to the candidate during the offer phase, in terms of timelines, selling our company to the candidate, etc; and if I know soon enough (the HR interview is usually first up in the morning, while mine is over lunch; I have other steps in between), I will also use that information to specifically address areas that I think we are a better (or worse; I try to be objective) fit for the candidate than the companies they're already interviewing with or have offers from.
What it does not affect is whether or not I will give an offer. I do that entirely based on our own interview process, as I know that other companies are often looking for different attributes in a candidate than I am, and we've spent a lot of time refining our process to select the folks that are well suited to the company, while maintaining good diversity of thought.
I also don't let it affect the offer $, though I would say there is likely a correlation due purely on the fact that folks who are coming to the interview with other offers in hand will probably interview well and have solid skills (they come in more relaxed and confident and that affects how well they perform).