Do I really need dedicated 20 amp circuit?
Code requires us to follow the MFG instructions per NEC 110.3.B .
Could there be issues using a circuit with multiple receptacles?
Possibly; additional splices or receptacles wired to code might create issues with devices that draw the maximum allowed amount if 1 screw or wirenut is not completely secure. This can create a high resistance connection where it may start arcing. Arcing can damage sensitive electronic components, so even without having anything else plugged in the multiple receptacles could be an issue, even if rare.
Because the instructions state it requires a dedicated circuit, a licensed electrician could not advise you otherwise.