Yes, received signals can form standing waves, and there is a very easy way for this to happen:
An object in the environment reflects some of your transmitted signal back to you.
You may have heard that antennas are influenced by their environment, and may need to be fine-tuned for that (rather than being exactly the shape that is specified by theory or simulation of an antenna in free space). This is exactly the same phenomenon, with a different description. Every such influence can be thought of as a partial reflection (in some direction, with some phase/polarization). An object separate from your antenna that is not fundamentally different from an element of your antenna — it's just usually farther away and thus has a lesser influence, but that's because we like to mount antennas in locations that maintain that separation and give predictable performance.
We can say that antennas are impedance matching devices, between your feed line and the environment. To have perfect SWR, an antenna must be adapted to the environment, and if the environment changes so will the SWR.
It might also be interesting to think about what happens when your antenna receives a signal that isn't identical to your transmitted signal. That will not be a standing wave, by definition. But, it is still traveling "in reverse" on your feed line, and your SWR meter will read it as "reflected" power because the meter doesn't care what the signal's frequency is, thus showing a SWR that is higher while not technically being due to a standing wave. This is usually insignificant because received signals are usually much weaker than transmitted signals, but we could hypothetically imagine pointing two nearby transmitters' antennas at each other (or just connecting them with a cable), in which case you've got possible damage to the final amplifier even though the signals need not be similar at all (no standing waves).
My point is that what matters, mostly, is not that the returning signal has the same frequency and phase (and so forms standing waves), but that there is a returning signal of significant power. Standing waves are just how we describe the condition in the feed line in the most practically relevant case, where there are reflections.
That said, standing waves in particular do have some effects:
- The theory of standing waves tells us what impedance the transmitter will observe, for a given length of line before the reflection. This could determine whether the transmitter will tolerate the reflections or not. (But in practice, we'd rather just avoid that case and the accompanying signal loss, hence why we care about low SWR.)
- Standing waves imply voltage and current peaks at stable locations on the feed line, which could overheat the line or break down insulation. (But in amateur applications, this is usually not a concern before the health of the transmitter.)