How to change pixel size of photo to 300 dpi on Photos app?

On a computer screen dpi means nothing. 72x72 looks exactly the same as 300x300. DPI means "dots per inch", and on a screen there is no "inch". dpi only comes into play when you print.

So to answer your question: it is not possible to change the dpi in Photos, not that it matters. Even when you do it in apps like Photoshop it's basically notional.

When you print it's a different matter. You have a finite number of pixels in your photograph. What size you print at decides the dpi. So, if you have a 24 megapixel image, say 6000 x 4000 pixels, and print it at 6 inches by 4 inches then the dpi will be 1000. Print it at 60 inches by 40 and the dpi will be 100. So it's in the print dialogue that you set the dpi. It's the only place that's relevant.

https://www.rideau-info.com/photos/mythdpi.html

I would also point out that 300dpi was once regarded a sort of minimum dpi for a good print. This is no longer the case. Printers have improved. So have cameras. The greatest determinant of a good print will be the quality of the printer and the sharpness of the exposure. You can get a perfectly good image these days at 100 dpi or a terrible one at 1000 dpi...

Nicely explained, Yer_Man. When you save or export a photo from Photos for Mac, only the dimensions (width and height and pixel) and the amount of compression matter. You can specify the dimensions and the compression in the "File > Export > Export ... photos" dialogue. To the far right of "Kind: PEG" is a toggle button, where you can reveal the advanced settings to specify the resolution and the amount of compression. If you export the photos using a lossy compression like JPEG or HEIC, you may want to use a high value for the quality and export at the full size to be able to print at a large size.