Which aspect ratio the image needs to be distorted to depends on whether or not it’s anamorphic widescreen, but, I think, in both cases, you’ll end up with non-square pixels, which makes proper display on LCD/LED panels difficult without something like interpolation. This is why using the Pixel-Perfect mode on the NES and SNES Classic makes games look horizontally squished.
As for your monitor, it sounds like it’s not really compatible with 240p or interlaced video modes, and a lack of proper aspect ratio controls is an unfortunate reality of cheap monitors (my Asus 22-inchers have zero aspect ratio control–full screen or bust–but my Dell U2410 gives me 5 or 6 options). If that 1080i is from a PS2, from something like Gran Turismo 4, I believe the internal rendering resolution is 540p, so squishing it back down to that probably looks okay.
Line 3x, 4x, and 5x for 240p default to 1280×720, 1280×960, and 1920×1080 frames respectively, which are all pretty standard PC monitor resolutions, so that may be why your monitor displays them correctly. 240p was never really accepted as a standard in terms of video equipment (it’s considered by some to be a perversion of 480i), and 720×480 is a EDTV thing and isn’t really seen in the PC world.