I'm wondering if you could tell me which sentence sounds more natural.
I'm going to have a job interview with a clothing company today.
I'm going to have a job interview at a clothing company today.
Both of them are syntactically correct, and I've heard both. However, there can be a slight amount of nuance when using with versus at. This is mostly due to cooperation and how much you feel like you're "on the same team" as the company. Just as you work with your colleagues, you can interview with a company. Depending on the magnitude of the job, saying that you're interviewing with a recognizable company by name might convey more importance than at.
In your example, you said "a clothing company," which, intentionally or not, removes some of the closeness in your context. If you were saying this verbatim, I would go with at (I'm going to have a job interview at a clothing company today) because you haven't given the name of the company directly. It doesn't give the closeness that with gives, but you've already distanced yourself slightly from the company by not giving its proper name.
I've also heard at used more in the context of a failed or disliked interview in past tense. For example:
Since the job wasn't attained, the implied closeness isn't needed. Many people want to distance themselves from unsuccessful attempts or careers that didn't speak to them.
All of this being said, however, I'd like to reiterate: both are technically correct and commonly heard. However, if you're trying to convey a tiny bit more information, you can do so by intentionally adding distance or removing it.
With X would be a person or company.
At X would be a place.
At X can also be used when X is a company, since many companies have offices, buildings, or campuses -- things that are places.
At X would never be a person, though.
At X also means the focus is more on where you are having the interview as opposed to who you are interviewing with. Since larger companies tend to have a more impersonal interview process where you might be waiting in a room, having to take significant time to park, etc. - the focus would be more on the place than your actual interviewer - thus at X could carry the implicaton you are interviewing with a large business or firm.