Light rays don't know the size of the sensor that they're headed for. So clearly the size of the format can't affect the 'rendering' or whatever other subjective property people want to attribute to MF. Imagine masking off the sides of a MF sensor with tape: you now have a full frame camera which must necessarily retain the 'medium format look'. Or similarly, imagine mounting an MF lens on a full frame camera (which you can in fact do). Aside from the crop, the rendering of the image must be the same.

The preceding considerations show that the medium format look, if it exists, must be a property that goes away as soon as you crop the photo. But none of the candidates for this alleged look (e.g. 'rendering' of out of focus regions) has that property.

In principle, medium format should offer shallower depth of field. However, in practice, medium format lenses rarely go beyond f2.8 whereas crazy ~f1 lenses are available for full frame. So if you really want ludicrously shallow depth of field you're probably better off with full frame.

All this being said, it's obviously the case that any given medium format lens might look different from any given full frame or crop sensor lens. Perhaps there are even generalisations about the 'look' of commonly used MF lenses vs. commonly used FF lenses (though I have never seen any evidence for this).

In-lens leaf shutters can also make a real difference, and are more common on medium format cameras.

On an anecdotal level, I'd add that I shoot a bit of 4x5 film as a hobby and the photos 'look' the same as the photos I take on my phone with its tiny sensor (apart from the expected differences such as resolution, grain, depth of field).

This site has some side-by-side comparisons of FF and MF: https://srussenschuck.com/review-fujifilm-gfx-50s/

To add to the sibling comment, it is real. But it's more about the field of view of the lens and the depth of field. As pointed out the light doesn't know the area of the sensor. Depth of field is a matter of the distance from the camera to the subject, the aperture of the lens and the focal length of the lens. Larger formats allow you to stand closer to the subject while retaining a wider view than for a smaller format camera. Take the 80mm f2.8 lens on a Rolleiflex TLR. If you stand where you can frame a head and shoulders portrait, you're standing closer than you would for the same framing on a smaller format camera with an 80mm lens. You would have to stand back further, with the subject further away more is in focus. Have a look at the Brenizer effect for a digitally stitched equivalent of a larger sensor.