The Strymon manual shows this signal flow diagram:
Thin lines indicate mono signals, think lines stereo signals. You can see that the whole delay circuit (starting with the record head) is being fed with a mono signal, that is created by summing (the “+” sign) the left and right input signal.
The individual outputs of each of the four tape playback heads are also mono each, and it is within the box labeled “stereo process” that the monaural signal of each playback heads can be individually panned. But each of this playback head signals is still a mono sum of the L & R input signals.
The “dry” path around the whole “tape head delay” is stereo, but I am unable to tell from the diagram whether or not the left and right input signals will be summed before being processed by sound on sound, reverb and spring.