Does exponential distribution assume proportionality?

Question

For example if I'm given $\lambda = 3$ births every $5$ minutes

this average rate must be proportional to the length of the period. Correct?

So $\lambda = 3/5$ births every minute via proportionality. Correct?

This is an assumption we must make to use exponential distribution just like we do with the Poisson distribution. correct? We must assume that the average rate they give us in a question is proportional to the length of the period. Is this right?

Answer 1

Something more fundamental is going on here.

$$ \frac{3 \text{ birth}}{5 \text{ minute}} = \frac{3}{5} \frac{\text{birth}}{\text{minute}} \text{.} $$

This is just the correct use of units (and remember that the use of singular or plural units is a convention -- the arightmetic of units does not care about the choice of this convention).