James Serra describes a few architectures for multi-tenant databases in the cloud:
Separate Servers\VMs
You create VMs for each tenant, essentially doing a “lift and shift” of the current on-premise solution. This provides the best isolation possible and it’s regularly done on-premises, but it’s also the one that doesn’t enable cutting costs, since each tenant has it’s own server, sql, license and so on. Sometimes this is the only allowable option if you have in your client contract that their data will be hardware-isolated from other clients. Some cons: table updates must be replicated across all the servers (i.e. updating reference tables), there is no resource sharing, and you need multiple backup strategies across all the servers.
Read on for a few other strategies. There aren’t any cloud-only details here; you could implement the same strategies on-premises.