Easy check. Pop the cover off. If the relay plate does not open and close rapidly while cranking the car, then there is a coil issue on the relay. Or, hold the relay closed manually (use something non-conductive, like a plastic pen) and see if the car turns on. When you release the pen from the plate and relay does not stay in the closed position, then that also confirms you have a power (potential) issue.

A relay is nothing but a switch. It works by a magnetic coil that pulls the contact plate closed. That magnetic coil need two potentials to be energized. If it does not close, you have a potential issue or the coil itself is compromised. Since this is a new relay (even though not OEM), I would sway from the coil itself being an issue. Or...the Autozone relay is crap or the incorrect one (high probability).