Restore Database on Active Passive Failover Cluster
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The SQL database will only need to be installed on 1 node, as the nodes share the storage already since SQL Server is 'cluster aware'.  However if you use something that is not 'cluster aware' such as SSIS, then packages will have to be installed on both nodes.

The reason you need to do 2 installations is because the nodes do not share installation files found on the local disk of the machine, but anything you do on shared storage and within SQL Server will be retained when failing over.  It's 1 database engine and 2 nodes.

Off topic, but a thought might be to have scripts that copy backup files and such so I make my 'backups' drive a cluster resource, and point all of my scripts to the virtual cluster address so even in case of a failover, they work.