Yep, it’s arising from altering the phase of a duplicated signal played alongside the original. You get it by passing the copy of the signal through a bank of all-pass filters modulated by an LFO, each of which affects different frequencies differently - so the phase of everything under, say, 500Hz would move at a different rate than frequencies between 100Hz and 1000Hz, and then differently than 1000Hz-5000Hz. Those differences is what produces the effect. The number of all-pass filters in a unit determines whether it’s a 4-stage or 8-stage or whatever.
A flanger is the same thing, except the phase shift is the same for all frequencies. That produces a comb filter that moves along the frequency spectrum. The signal is then fed back through the circuit to create resonances.
Chorus works a bit like a flanger without the feedback, except the delay between signals is longer. You start with the original signal, pass it through an LFO-modulated pitch shifter or two, and then delay that signal alongside the original.
Reverb is just a series of delays, which is what actual reverb is - sound bouncing off things and back to your ears. If you clap in a very, very large space, it takes awhile for the sound to bounce off the far wall and get back to you, long enough that your brain interprets it as a separate, discrete event. In a smaller space, you get the direct hit of the original wave, and then shortly after all the weaker waves that have bounced off walls and cars and dogs and hot dogs come back and hit your ears, fast enough that your brain sort of smears everything together into one reverberated sound.
To simulate that, reverbs just string a bunch of delays together over a very short time and blend them via various methods, usually with some filtering. Most digital multi-tap delays can be set low enough that you can get a reverb out of them if you fiddle enough.