If we multiply a principal amount $A_0$ in dollars (\$) by an interest rate $r$ in percentage (%) once, let's say for a month (m), then we have an amount $A$ in dollars per month ($/m)
$$ A(\textrm{\$/m}) = A_0(\textrm{\$}) \cdot r(\textrm{%}) $$
The problem is that for the units to agree arithmetically, the interest rate would have to be in units of per month (/m). This is also the case if we multiply the interest rate by a period of time $t$ in months per year (m/y), then we have an amount in dollars per year ($/y)
$$ A(\textrm{\$/y}) = A_0(\textrm{\$}) \cdot r(\textrm{%}) \cdot t(\textrm{m/y}) $$
Once again, the interest rate would have to be in units of per month (/m) which looks strange. This is still the case for the instantaneous rate of change of the amount with respect to time $\frac{dA}{dt}$ in dollars per period of time ($/m), which looks a lot like the formula for exponential growth and decay
$$ \frac{dA}{dt} (\textrm{\$/m}) = A_0 (\textrm{\$}) \cdot r(\textrm{%}) $$
Yet again, the interest rate would have to be in units of per month(/m). I've always considered a rate like a scaling factor without thinking much of the units, so how can I make sense of the units for the above interest rate?
Your dimensions are correct, the proper interpretation to your interest rate is that it's the amount of money that you earn per month, scaled by the principal amount, so it simply has units of "per month". If you like, you could view it as a frequency (related to the frequency of doubling your investment etc.)
In general, a derivative $ \frac{d}{dx}$ acts as if it has dimensions of $\frac{1}{x}$ - e.g. a velocity has units $m/s$ and is given by $v = \frac{dx}{dt}$ where x has units of $m$ and t has units of $s$. More complicated derivatives (such as partial derivatives, divergence, etc) also follow this pattern.