Why does reflection only applies to transmission lines?
Reflections happen everywhere, not just in transmission lines. Transmission line is a model of the physical situation, which is easy to apply to a pair of conductors whose length is comparable to or larger than the wavelength of the signal, and which is regular in cross section.

What determines whether reflections matter is the frequencies in and the physical size of the circuit. If you have unmatched impedances then you do get reflected waves just as you describe, and either you have to deal with them or they are negligible for some reason. Here are two reasons:


For exclusively low-frequency circuits, the reflections reflect repeatedly and settle down on a timescale much faster than the signals change. That is, each double reflection is an extra signal which is merely out of phase with the original signal, but as they get more out of phase their amplitude drops quickly enough that they can be neglected. (Even RF circuits can be built this way, as can be seen from a lot of homebuilt HF amateur radio gear.)

As frequency increases, wavelength decreases, and the physical size of your components becomes relatively larger, and you start having to worry about avoiding impedance “bumps”. This is where you start using microstrip design techniques in printed circuits.
In digital circuits, sharp transitions may have high-frequency components that will reflect but you don't have to worry about this as long as your clock speed is much slower than the length of your traces/wires (there's a conversion via c to make that make sense, of course) because by the time the clock makes its next tick all the signals have settled down to a steady state.

(Note that there are no standing waves here because within the period of a single clock tick the driving signals are steps (high to low or low to high logic levels), not periodic signals.)

As clock speed increases, the settling time available decreases, requiring you to either minimize reflections or minimize signal travel time (so that the settling occurs faster).