No. The exact pump must be designed for that fixture and its wastewater conditions. A spare inlet does not establish suitability.
What the observation means
Temperature, flow and waste composition differ from toilet use. Existing connections may need assessment rather than being assumed correct.
A useful next step
Identify the unit and describe all connected appliances when seeking help. Do not add or reroute a washing-machine hose yourself as a workaround for another drain. New installation work is separate from repair of an existing accepted arrangement.
Manuals and supporting guidance
Model-specific instructions take priority. These references support the practical principles discussed here; they do not diagnose your appliance.