Like units can be added together or, subtracted from one another. However, multiplication and division of units does not have such boundations. multiplication is just repeated addition, similarly division is repeated subtraction. How in the world we don't have same conditions as that on addition of like units?
If you mix (add) $5$ oranges and $2$ cars, you still get $5$ oranges and $2$ cars.
However, if you mix (add) $5$ oranges and $2$ oranges, you can compute the sum and we say that we get $7$ oranges.
The point is: to add and subtract, you need to have the same type of "things".
These examples use integer arithmetic, since it is a concept that we can visualise. However it would be easy to expand to continuous measures like:
Mixing (adding) $0.5 \ kg$ of sugar with $0.2 \ kg$ of sugar gives $0.7 \ kg$ of sugar; while mixing $0.5 \ kg$ of sugar with a ruler $0.2 \ m$ long gives the $0.5 \ kg$ of sugar and a ruler $0.2 \ m$ long.
With respect to multiplication, the multiplication can be thought as having $2$ boxes with $5$ oranges each, which results in:
$2$ box(es) $\times 5$ oranges/box = $10$ oranges. Note that oranges/box can be read as "oranges per box".
I tried to give a simple answer to your question. A more complex answer could lead us to dimensional analysis.