Multiple Databases Vs Single Database with logically partitioned data

You'll wish you had used separate databases:

• If you ever want to grant permissions to the databases themselves to clients or superusers.
• If you ever want to restore just one client's database without affecting the data of the others.
• If there are regulatory concerns governing your data and data breaches, and you belatedly discover that these regulations can only be met by having separate databases. (Update: a little over 4 years after the writing of this answer, GDPR went into effect)
• If you ever want to easily move your customer data to multiple database servers or otherwise scale out, or move larger/more important customers to different hardware. In a different part of the world.
• If you ever want to easily archive and decommission old customer data.
• If your customers care about their data being siloed, and they find out that you did otherwise.
• If your data is subpoenaed and it's hard to extract just one customer's data, or the subpoena is overly broad and you have to produce the entire database instead of just the data for the one client.
• When you forget to maintain vigilance and just one query slips through that didn't include AND CustomerID = @CustomerID. Hint: use a scripted permissions tool, or schemas, or wrap all tables with views that include WHERE CustomerID = SomeUserReturningFunction(), or some combination of these.
• When you get permissions wrong at the application level and customer data is exposed to the wrong customer.
• When you want to have different levels of backup and recovery protection for different clients.
• Once you realize that building an infrastructure to create, provision, configure, deploy, and otherwise spin up/down new databases is worth the investment because it forces you to get good at it.
• When you didn't allow for the possibility of some class of people needing access to multiple customers' data, and you need a layer of abstraction on top of Customer because WHERE CustomerID = @CustomerID won't cut it now.
• When hackers target your sites or systems, and you made it easy for them to get all the data of all your customers in one fell swoop after getting admin credentials in just one database.
• When your database backup takes 5 hours to run and then fails.
• When you have to get the Enterprise edition of your DBMS so you can make compressed backups so that copying the backup file over the network takes less than 5 hours more.
• When you have to restore the entire database every day to a test server which takes 5 hours, and run validation scripts that take 2 hours to complete.
• When only a few of your customers need replication and you have to apply it to all of your customers instead of just those few.
• When you want to take on a government customer and find out that they require you to use a separate server and database, but your ecosystem was built around a single server and database and it's just too hard or will take too long to change.

You'll be glad you used separate databases:

• When a pilot rollout to one customer completely explodes and the other 999 customers are completely unaffected. And you can restore from backup to fix the problem.
• When one of your database backups fails and you can fix just that one in 25 minutes instead of starting the entire 10-hour process over again.

You'll wish you had used a single database:

• When you discover a bug that affects all 1000 clients and deploying the fix to 1000 databases is hard.
• When you get permissions wrong at the database level and customer data is exposed to the wrong customer.
• When you didn't allow for the possibility of some class of people needing access to a subset of all the databases (perhaps two customers merge).
• When you didn't think how hard it would be to merge two different databases of data.
• When you've merged two different databases of data and realize one was the wrong one, and you didn't plan for recovering from this scenario.
• When you try to grow past 32,767 customers/databases on a single server and find out that this is the maximum in SQL Server 2012.
• When you realize that managing 1,000+ databases is a bigger nightmare than you ever imagined.
• When you realize that you can't onboard a new customer just by adding some data in a table, and you have to run a bunch of scary and complicated scripts to create, populate, and set permissions on a new database.
• When you have to run 1000 database backups every day, make sure they all succeed, copy them over the network, restore them all to a test database, and run validation scripts on each single one, reporting any failures in a way that will guaranteed to be seen, and which are easily and quickly actionable. And then 150 of these fail in various places and have to be fixed one at a time.
• When you find out you have to set up replication for 1000 databases.

Just because I listed more reasons for one doesn't mean it is better.

Some readers may get value from MSDN: Multi-Tenant Data Architecture. Or perhaps SaaS Tenancy App Design Patterns. Or even Developing Multi-tenant Applications for the Cloud, 3rd Edition

If you are refering your architecural as "multi tenant", Microsoft has a good article which is worth to read here. It shows some comparison between "isolated" (multiple db) and "shared" (single db). Generally, shared wins when the # of tenant (client) is big, but when the size of each tenant is big, an isolated approach is recommended.

Those consideration however can only be calculated by experienced developers though.

Still if you managed to use isolated (multiple db) architecture, you still won't get direct benefit in performance when they are still run at same instance. And if you use shared (single db) architecture, consider using int instead of guid, or sequential guid if you still need to use it.