It is possible to use app keys for all the connection string inputs and read those on connection string like bellow
<add name="DefaultConnection" connectionString="Data Source=$(Server);Initial Catalog=$(Catalog);User ID=$(User);Password=$(Password)" providerName="System.Data.SqlClient" />
<add key="$(Server)" value="xxxx" />
<add key="$(Catalog)" value="xxxx" />
<add key="$(User)" value="xxxx" />
<add key="$(Password)" value="xxxx" />
As @Ertürk Öztürk already say - it's not possible.
If you searching for more or less clean way to do it i suggest you to use SqlConnectionStringBuilder or DbConnectionStringBuilder if you using not MSSQL data base.
In your code it will be like this with SqlConnectionStringBuilder:
//create connection string builder
System.Data.SqlClient.SqlConnectionStringBuilder connectionStringBuilder = new System.Data.SqlClient.SqlConnectionStringBuilder();
//set all properties from your WebConfig
connectionStringBuilder.DataSource = ConfigurationManager.AppSettings["Server"];
connectionStringBuilder.InitialCatalog = ConfigurationManager.AppSettings["Catalog"];
connectionStringBuilder.UserID = ConfigurationManager.AppSettings["User"];
connectionStringBuilder.Password = ConfigurationManager.AppSettings["Password"];
//not you can get rigth formatted connection string
var connectionString = connectionStringBuilder.ConnectionString;
It's not possible. Actually you don't need to do this, that's why it's not possible. Because you can change the other parts of web.config same like AppSettings.
ConfigurationManager.ConnectionStrings["DefaultConnection"].ConnectionString =
String.Format("Data Source={0};Initial Catalog={1};UserID={2};Password={3}",
"server", "db", "ID", "Pass");
$(server)and other settings individually elsewhere in the application as well.