If I am connected to a Home Pod, can my wife also use siri on the Home Pod that I am connected to?

Siri on HomePod doesn't distinguish between voices. It will attempt to execute any command it recognises. If set up, personal requests relating to you calendar, tasks, voice mail, etc. only work when the iOS device you used to configure the HomePod is on the same network.

tt2

She can only use the Siri from your icloud account. So, she can troll you... :-)