The one database per user solution just wont scale!
Say your system needs 20 tables, then you start hitting magic (or more correctly cursed) numbers like 32760 with only 1638 users. Plus most admin tools will slow to a crawl when your system contains thousands of tables, indexes etc. And you backup and maintenance becomes a full time job.
To use a single database you only need to add one extra "owner" column in each table and index.
Also depending on what your "SaaS" service actually is I doubt you will end up with a simple "one user uses only one set of data, and, one set of data is used only by one user".
Think of familys and small companies where access to the system is shared by several users. OK they could just share the user id but they would probably like some sort of audit trail as to who exactly did what, and, perhaps different levels of access to the data.
You also have the opposite sistuation where a single user may want to adminster several sets of data (say a church member adminstering a set of data for the sunday school, the soup kitchen and the roof repiar fund).
The normal practice is to associate the data with a "contract" and to give individual users various levels of access to the contract or contracts.