Yes. Any difference is a difference that requires you to make the choice between fitting options like fill frame proportionally, which will crop the image on the dimension that is too large, fit to frame proportionally, which will leave some empty space in the frame, but not crop anything, or fit to frame and distort your image, which is a very poor choice.
You can set up content fitting with the object styles. By applying a different object style it will apply the fitting if activated as option in the object style. Problem is only that a different style must be applied so therefore I have often 2 styles with fitting options, to apply another one first, and the correct one as second step.
If the frame has applied such an object style with fitting features, any image wich is dragged into that frame will fit automaticaly.
There are different fittings, fill or fit, proportionally or disproportionally. If the image has a different ratio than the frame, you have either to mask the image with the frame or to not fill the frame completely or to fit disproportional. But in my opinion you need not to rescale it in Photoshop.
I tried this, and sometimes it worked and sometimes it didn't.
There's math in page layout, and one place it comes into play is fitting an image into a frame, as you have discovered. One place it comes up is whenever the aspect ratio of the frame is different from that of the file within the frame.
Unwilling to think too much about the math, here's how I teach this in class: if the frame size is a set size and you want the picture to fit inside, try fitting button #1 (Fill Frame Proportionately) and then button # 2 (Fit Content Proportionately). One will be better than the other. Pick the one you like best and move on.
As per Peter's comment, there are very few valid reasons to purposefully distort an image, so avoid button #3 which is lacking the word "proportional". It will slightly or wildly distort an image that has a different aspect ratio than the frame that contains it.
Should I just go back to Photoshop and resize my photo to 6" x 5.25?
You would have to choose between a distortion and a crop there also—the aspect ratios don't match