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. – StephenG Feb 3 '18 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: