Is the percentage symbol a constant? 
*

*Isn't the percentage symbol actually just a constant with the value $0.01$? As in 
$$
15\% = 15 \times \% = 15 \times 0.01 = 0.15.
$$

*Isn't every unit  actually just a constant? But why do we treat them in such a special way then?

If you want to think about units or dimensional analysis, then probably it is best to interpret 15% as
$$ 15\% = \frac{\text{$15$ units of X}}{\text{$100$ units of X}} $$
(I would read this as "15 units of X per 100 units of X".) For example, the Wikipedia page on dimensional analysis gives the example of debt-to-GDP ratio.
  $$ 90\% \text{ debt-to-GDP} = \frac{\text{$90$ dollars of debt}}{\text{per $100$ dollars of GDP}} $$
Here both the top and bottom are dollars. But they are two different dollar measurements. So even though the percentage is a dimensionless quantity (dollars/dollars), keeping the units in mind may be wise. Similarly
$$ 10\% \text{ full} = \frac{\text{$10$ liters of water}}{\text{per $100$ liters of container}} . $$
and so on. (edit) So for example, if you want to do a calculation like "How many liters of water are in a 2-gallon container that is 10% full", you do
$$ \text{$2$ gallons of container} ~ \cdot ~ \frac{\text{$3.78$ liters of container}}{\text{$1$ gallon of container}} ~ \cdot ~ \frac{\text{$10$ liters of water}}{\text{$100$ liters of container}} . $$
Of course you could have also done
$$ \text{$2$ gallons of container} ~ \cdot ~ \frac{\text{$10$ gallons of water}}{\text{$100$ gallons of container}} ~ \cdot ~ \frac{\text{$3.78$ liters of water}}{\text{$1$ gallon of water}} ~ \cdot . $$
