I don't know how it's done, but these are the possible scenarios:
- DB-per-customer
- schema-per-customer
- single-schema
Case 1 is trivial in terms of backup/restore (or import/export), case 2 is similar. I would venture a guess that these 2 are the most used approaches.
The third option makes export/import difficult, but not impossible. The basic idea is that a table holds data from all companies, but distinguishes the company by a foreign key. Export and import would require the same kind of ETL tool to be used because these actions require filtering by company ID. The export procedure takes a company as a parameter and runs the task for that company only. The dump would take the form of insert statements (like the one you can get with MySQL or PostgreSQL) or XML (like the one created by DDLUtils).
There are situations where the single-schema setup comes in handy, but I don't think multi-tenancy is one of them.