Hi actually we deploy code as setup file (.msi), you can create setup project using Wix or install-shield ( maybe other too,but these two are most popular and widely used)
You can do database deployment in two ways
1.) you can provide SQL script for generating SQL server and give instruction for generating Database from Script. It is simple and easy to do.
2.) Other way is provide option to create database from setup. It need some work, but it is more good way.
Now come to your question, you can provide some UI in setup that take input from user and connect to appropriate DB. Or create a UI for Updating configurations, and save configuration (connection string ) in some file ( ex: .ini or .config file).
Below are the some URL for creating Setup and connecting to Db using Wix:
WIXDataBase
installing-databases-using-wix
Creating-an-installer-using-Wix
using-wix-to-install-sql-databases-and-execute-sql-scripts
WiX Samples