Can more than one trading account be opened with Zerodha?

No, only one trading account can be opened with one PAN with one broker. However, a new non-individual trading and demat account, in the name of a company, partnership firm, LLP, or HUF can be opened even if one holds an individual account with the same broker. To open a non-individual account, see How to open a non-Individual account with Zerodha?