I think you would have to convert the SQL statements where you have client name into dynamic SQL statements and execute them to achieve this. For example, if your create statement had client name in it, you would change it to use a parameter like shown below.
DECLARE @clientName NVARCHAR(64) = 'Google';
DECLARE @sql NVARCHAR(4000);
SET @sql =
'CREATE DATABASE '+QUOTENAME(@clientName)+' ON PRIMARY
( NAME = N'''+@clientName+ ''', FILENAME = N''F:\Data\' + @clientName + '.mdf'' , SIZE = 3072KB , FILEGROWTH = 1024KB )
( NAME = N'''+@clientName+ '_log'', FILENAME = N''G:\Logs\' + @clientName + '_log.ldf'' , SIZE = 1024KB , FILEGROWTH = 10%)';
exec sp_executesql @sql;
If these databases are on separate servers, it would be advisable to use the same database name. That will make maintenance, access, coding etc. a lot simpler.