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It doesn’t make sense to create a new database for each organization. Even if the number of customers or organizations grows to the hundreds or thousands, keeping data in a single database is your best option.
Edit:
Your original concern was that an increase in the number of organizations would impact performance. Well then, imagine if you had to create an entire database for each organization. You would have the same tables, indexes, views, etc replicated n-times. Each database would need system resources and disk space. Not to mention the code needed to make Django aware of the split. You would have to get data from n databases instead of just one table in one database.
Also keep in mind that robust databases like PostgreSQL or MySQL are very capable of managing thousands or millions of records. They will do a better job out of the box than any code you may come up with. Just trust them with the data and if you notice a performance decline then you can find plenty of tips online to optimize them.