Has our new septic tank been properly waterproofed?
Probably not.

I have a precast cement tank. It is not otherwise waterproofed/coated, etc. It's perfectly standard to use them like that. They don't spray water everywhere - it simply seeps, rather slowly, through the walls.

Consider, if you will, what happens to water that you keep in the septic tank. It sits for a while, and then leaves by a pipe whereupon it's diffused in the soil in the drainage field.

Any seepage through the cement/mortar will be quite slow, and will also diffuse water into the soil. Sure, it will be the soil around the tank, but there's no significant difference. 

If you feel the need of an official opinion, call the local health department (whoever does your septic permits/inspections) and ask for an official opinion.