DopeGhoti's answer (No!) is the correct one, as long as we're talking standard software, which means reading the passphrase as a single line of text which you can type any way you want (and even correct typos), until you hit Enter.
That said, it's technically possible to distinguish Left/Right shift keys, as
xev easily demonstrates. So as long as the keyboard itself sends different keycodes depending on which key was pressed, you could work with it - if you go to the trouble of creating your own wrapper that reads individual keystrokes instead of text.
Going further, you could even add things like having to enter the passphrase in a specific rhythm, or whatever. Kind of like cheatcodes some old games used to have. You could even use a real game controller to input those...
So (Yes!) almost anything is possible, but you'd have to develop a program that provides such functionality, and make it turn your desired input into a textual representation or hash, which can then be used as the passphrase. And the problem then is to make it such that the user has a realistic chance of re-producing the same representation/hash repeatedly.
So if you do decide to turn your passphrase input into a rhythm game, it should allow some generous leeway. With a regular keyboard pressing shift twice (without typing a letter) is no harm done, how would your software deal with such misinputs?
In any case, with effort, it's possible. The bigger question then, is whether there is any point in doing this. Distinguishing left/right shift key only adds 1 bit of entropy anyway. It doesn't seem very useful.
Passphrase wrappers in general are not unheard of, it's done to support dedicated hardware for example (like integrating a yubikey challenge response). 2-factor authentication, high entropy passphrase, may be worth it.