If "thinking" affects feeling, and "feeling" affects thinking, then "why" does this not produce a never-ending circle?

For the same reason perpetual-motion machines can’t work: the first and second laws of thermodynamics. Energy is lost from the system, and that energy can’t magically regenerate. Put colloquially, we exhaust ourselves after a while. Another way of looking at it is that, at some point, you get distracted or, worst-case scenario, fall asleep.

Because (sleep aside) this happens eventually:

angry feelings -> angry thoughts -> angry feelings -> realization thoughts -> some other kind of feelings

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But in a sense it is a never-ending cycle. But it’s not

There’s a never-ending cycle (at least while we’re awake) of mental-event A causing mental-event B, causing mental-event-C and so on, but various realizations and distractions intrude, changing the course of the cycle. So it’s more of a chain than a cycle.

The thought-feeling loops you mentioned are real, even if they don’t last forever. They can be big problems and last for a long time or happen frequently. One of the main reasons people go to therapy and/or take psychedelics (as medicine) is to break out of these loops. Either the therapist or the drug intervenes, essentially shaking the snow-globe of the mind up, making its snow fall in different patterns.

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What a feeling (and emotion) really is, is a body sensation plus a thought. So feelings themselves are partly made of thoughts. One way to see that this is true is to note that the body sensations of being anxious (e.g. about a big test coming up) and the body sensations of being excited (e.g. about an exciting movie) are almost identical. What makes them different feelings is a thought context, tagging them as bad and good, respectively.

… forever.

All of the above is a toy version of human psychology. It’s not really a sequence of thought, feeling, thought, feeling, thought, feeling … It’s much more complex and messy than that, with thoughts and feelings intertwined. Plus, a thought can lead to another thought, instead of a feeling, and a feeling can lead to another feeling, instead of a thought. Or a feeling can lead to a thought and a feeling, etc.

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Those realization thoughts could be anything that, in some way, changes the feelings. An example is a memory that pops up, in which the person you’re angry at gave you an amazing birthday present.

angry feelings -> angry thoughts -> angry feelings -> distraction -> some different kind of thought or feeling.

Which is why so many therapies and meditation practices are about detaching thoughts from body sensations, learning to seem them as two separate things, rather than as a conglomeration.

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Also consider an itch. It probably doesn’t trigger much of anything on its own—except maybe slight irritation. It’s just a body sensation. But try coupling that with the thought of “skin cancer.” Your state of mind will be very different, but the sensation will be the same.

angry feelings -> angry thoughts -> angry feelings -> angry thoughts

This can happen:

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E.g. the person you’re angry at suddenly drops his pants. Thought that might make you even angrier. Who knows? Still, eventually there will be some sort of distraction. (For some people, sadly, it’s alcohol or some other unhealthy or addictive thing.)