Does the brain process sound in "clips" or "frames"?
In your question, you ask about processing and sampling. I'd like to make a distinction between sampling, which in my view does not have a framerate, and perception which there is some evidence that there is sort of a framerate (though not as concrete as the framerate of a video display, for example). Processing is everything in between, and as such, whether it shows "framerate-like" behavior depends on the stage of processing.

Does vision really have a framerate?

The "framerate" depicted in those sources for vision is sort of a special case, talking about perception of a particular stimulus. There are no real "frames" in your vision, only temporal averaging (because phototransduction is quite slow on the timescale of neural processing in general). The article you link makes a mistake in logic in concluding that humans have a frame rate: the research they cite is about discriminating between separate flashes, which is an issue of temporal averaging rather than framerate. There is a separate issue, which is whether perception correlates with particular oscillatory frequencies; I'll get to this later in my answer.

If you flash a very bright 1 ms light, and record it with a video camera, you will often miss the light entirely (or it may appear only in part of the image, depending on the scanning pattern of the camera). With vision, though, you will always see the light, and for very brief flashes you will perceive changes in duration as changes in intensity.

The other contributor is the "magic" of perception: effectively, what you perceive is not directly the signals that come in through your eyes but rather the model your brain makes for the world based on that information. This process is the origin of all perceptual illusions; the appearance of "backward" images like with a spinning wheel is one example. 

Our current model for how processing of sensory stimuli works in the brain is that there is a functional hierarchy with significant abstraction at higher levels of the hierarchy. As you get to higher order levels of processing, you will cease to see the "pixel information" and simple edge/feature detection, and begin to see more abstract representations like detection of particular visual objects or encoding of object velocity.

Okay but the question is about auditory stuff..

It's easiest to start with vision simply because, for better or worse, it is best studied historically. For auditory processing, there are a lot of similarities with vision even though the input stimuli start very different. At low levels of processing, auditory neurons are well described by spectro-temporal receptive fields. At higher levels of processing, you will instead see representations of whole features in sounds, such as phonemes.

Object segmentation is a very difficult problem in both vision and hearing (and a difficult task for computer/AI-based versions of those senses, as well). This is definitely an active field of research, but there is good evidence that some segmentation is present throughout the cortical hierarchy (for example, see here). There are no concrete answers for 'how' exactly segmentation works, just that it does, some evidence for where it might be happening, and some understanding of factors that make it harder to segment different sources. Sound localization is certainly one of the key factors, though segmentation can occur even when there are no localization cues. For example, if you have multiple speakers on the radio, or multiple musical instruments playing, there are some shared spectral characteristics of the speakers or instruments

Discrete signals in cortex

All that said, this does not mean that the brain is free from any discrete elements of processing (like "frames"), although it probably doesn't make sense to think of a frame rate but rather a type of "threading". There is recent strong evidence that activity in the neocortex can be organized into packets of activity that contain sensory or other information. So even though the auditory stream is sampled continuously, that continuous sample is most likely distilled into discrete auditory objects that reflect both the content and source of those stimuli.