It seems that there is some colour management done by the printer driver even if you specify "no colour management".
Printing the exact same ink onto different papers will give different results. On the other hand, if you tell the printer you have a different kind of paper in the printer it will alter the inks to match the paper. The reason the prints look different is because the printer is changing the ratio of the inks to match a different type of paper when you didn't change the paper.
That is why the printer dialogue is asking what type of paper is loaded - so it can use the appropriate blend of inks for each specific color in order to make the image look the way it should based on the type of paper loaded in the printer. If you tell the printer you have one type of paper in it when you really have another type, you won't get accurate colors.
The different stages in your workflow are like different links in a chain. Photoshop manages color while you are processing and editing using Photoshop. The output from photoshop is in a standard graphics format (JPEG, TIFF, PNG, etc.). The printer driver has to translate what PS sends to it in order to print it as accurately as possible. In order to do that it needs to know what kind of paper is being used so it can adjust the inks to match the paper and give you a print that looks like what came out of PS.
When you select a paper type from an option within PS, it is attempting to access the printer driver and change it for you. That change may or may not be reflected when you subsequently open the printer dialogue (because the printer dialogue may default back to a specific setting when you open it), but if you print the image without subsequently opening the printer dialogue the paper choice selected by PS should be applied to the print job.