I use Inkscape and want to export a PNG image (of my SVG file) with dimensions 1080 px by 1920 px. However the DPI is always set to 90 and if changed, the dimensions change with it. Is it possible to have a higher DPI with the same dimensions?
1) If you want a file 1920x1080 px, the ppi is TOTALLY irrelevant, because you want it for an electronic medium.
2) In the case of the export focus on the pixel size. Yes, on the moment of the export, the ppi and the dimensions are intertwined but after that, you can change that in gimp or another aplication like IrfanView.
3) You could try to change the dimensions before exporting, but there is no point in doing so.
No, you can't.
In Inkscape, DPI box on the Export window refers to actual pixels. and you can't decouple it from pixel size. The fact that the option is on the export window has always confused me (and probably lots of other people).
It looks like you can get the results you are looking for by increasing the document size, but I wouldn't bother.
What you want to do is export to the desired pixel size, and ignore the DPI setting in Inkscape. After you've exported, open up the image in a Raster editor, like Gimp, and change the pixel size. In Gimp, the option is under Image > Print Size.
As others have pointed out, the DPI is just metadata.