If you cut a piece of tape as a measure of the width of your computer monitor, then to tell some one else remote from you this width you either have to physically send them the piece of tape or you have to compare the length of the piece of tape with a standard of length that you can both agree on e.g. the metre. Once you have both agreed on a common standard you can communicate a dimensioned number to tell them the screen width. The act of measurement is basically a ratio comparison of two quantities of the same type e.g. two lengths. If one of them is a standard length or some known proportion of a standard length, then the measured/calculated ratio has the dimension of that standard of length.

For convenience you will normally use a tape pre-marked out in standard units so the width of the monitor can be measured simply and directly, without thinking deeply about the underlying measurement process.

If you calculate the ratio of two lengths measured using the same standard then this ratio becomes dimensionless.

If you calculate the ratio of two lengths measured using different standards then this ratio becomes a conversion factor between the two different standards.

There are some oddities. e.g. energy and torque have the same units in the metric system, i.e. Nm, but they are physically different types of measured quantity.