What allows us to treat physical units in algebra?

$$Speed = \frac{Distance}{Time}$$

Following this, is makes sense that the units of speed is m/s. However, I do not follow why we are able to divide units to derive a new unit, which turns out to be intuitive later on (e.g. meter/second gives you 'meter per second' which makes intuitive sense as a unit of speed).

• You are over-thinking this. The unit follows from the definition of speed. They don't "turn out to be intuitive later on", they start out as being just as intuitive as the definition. Feb 3, 2018 at 5:23

You are confusing dimensions and units. Dimensions group together co-measurable units. For example 5 seconds from now (second is a unit of measure) is $\frac{1}{12}$ minutes from now where the conversion made is multiplication by dimensionless quantity.
Another example: In Special relativity due to nature of $c$ and/or the identification (up to an important sign) of spacial and temporal units having the same dimension renders velocity dimensionless: