I am creating a new Google Apps Script projects for clients of mine. These projects are only owned and managed by my single Google Account. These Apps Script projects are duplicates of each other with changes in document IDs. I have a total of 6 Apps Script projects, and I am looking to add 20 more clients (so 20 more projects).
Will this slow down the execution time on my end as well as my clients end? Does each project run on different servers? Does the amount of project on 1 Google Account play a role in execution speed?

As pointed out by @Alan Wells the execution time is not affected by the amount of projects in your account, however your Apps Script quota limits apply to the sum of all your projects.
And if the code were a running as an Add-On it would probably do so on different servers as the users would be running the code from their account, and likely in different geographic locations.
But to summarize: