Lots of databases on one server

I have around 1000 websites to be hosted, All websites are hosted on the same server(LAMP), each website has its own database (DB_1, DB_2,… DB_999,DB_1000)

Each database has around 50 tables, All the websites have less traffic…

  1. Does hosting too many databases on the same mysql server slow it down?

  2. Which is preferred? hosting 1000 databases with 50 tables each or hosting 1 database with 50,000 tables?

No, generally a lot of DB does not affect performance.
The total size of data, queries per second and if you have proper indexes are much more important.

1000 DB 50 tables each.
Due to OS and which file system you are using, having a lot of files in one directory is expensive.
And if you have 50,000 MyISAM tables in one database that will physically mean that you have 150,000 files in one directory.
If you have change to InnoDB that figure will drop to 50,000 which is still too high.