Does a higher-megapixel camera magnify the subject or just record more detail?

No, more megapixels does not magnify the image (meaning the subject image). Focal length (zoom) magnifies the subject size in the image. Magnify is a word referencing the subject size within the image frame. Magnification is increased by longer focal length and/or by closer subject distance.

More megapixels is a larger digital image (if measured in pixels), but that is not magnification, as it does not change the lens view of the subject. More pixels can improve the digital sampling reproduction of the lens detail. And more enlargement may see that lens detail larger and see small detail more clearly, but neither can increase the lens detail in the image. Magnification is a longer lens or closer distance that does increase subject size in the sensor frame, allowing the same pixel resolution of finer detail now seen.

More megapixels does Not increase the resolution of the lens, but it likely does increase the reproduced resolution of the viewed detail present, seeable if you enlarge it to see it, and more megapixels does allow greater viewing enlargement of that detail. But the lens resolution is still the absolute limit on image detail. Only the lens creates image detail, digital sampling can only try to reproduce what is there.

I'm copying my comment below up to here, with improved wording.

Cropping an image smaller and then necessarily enlarging more (to be the same viewing size again) does superficially "look" exactly like zooming. This is just an illusion, and my site at https://www.scantips.com/lights/cropfactor.html has elaboration about exactly that illusion (in regard to crop factor).

The cost in lost pixels of later cropping any finished image greatly lowers the resolution of any enlargement.

Regarding crop factor on the sensor, the cropped image is simply a smaller image (in terms of sensor size in mm). It may have the same megapixels as a full frame image, but the necessary greater enlargement factor of any smaller cropped image lowers the viewed resolution (and viewing is the only goal). It "spreads" the pixels further apart, showing fewer pixels per inch.

Some of us ignore that, and do imagine a cropped sensor magnifies using a shorter lens, but it simply is not magnification. It may be very adequate in most cases, but in the most critical case, it is just detrimental greater enlargement of a smaller image and a shorter lens. There is no other way to say it.

Magnification routinely refers to actual lens magnification.