[Feature Request]: Database name in Connection Dialog should be a drop down

### Feature Description

The Database name property in the Connection Dialog should be a drop down of all the available databases on the server. This makes it much easier than needing to remember exact database name.

### Problem and Motivation

1. Connect to a server/database

Expected: Populate connection details for server and database name is auto populated, similar to other client tools.

### Related Area

- [x] Connection dialog (SQL Server | Azure browse/Fabric browse)
- [ ] Query editor
- [ ] Query results panel
- [ ] Object Explorer
- [ ] GitHub Copilot integration
- [ ] Preview/Edit data
- [ ] Table Designer
- [ ] Schema Designer
- [ ] Schema Compare
- [ ] Local SQL Server Container provisioning
- [ ] SQL database in Fabric provisioning
- [ ] DACPAC/BACPAC export/import
- [ ] SQL Database projects
- [ ] Query Plan Visualizer
- [ ] Other (please describe below)

### If you selected "Other", please describe the area

_No response_

### Confirmation

- [x] I have searched existing feature requests and couldn't find a match
- [ ] I want to help implement this feature

Start by locating the TypeScript implementation of the Connection dialog for SQL Server and Azure browse/Fabric browse, then trace how server and database connection details are populated. The work is complete when the Database name field offers all databases available on the connected server and can populate the selected database.

Written by the indexing model from the issue text.