Reference same Connection String in Multiple .NET Applications

Microsoft has an amazing documentation about it. Hopefully that helps.

To store connection strings in an external configuration file, create a separate file that contains only the connectionStrings section. Do not include any additional elements, sections, or attributes. This example shows the syntax for an external configuration file.

<connectionStrings>  
  <add name="Name"   
   providerName="System.Data.ProviderName"   
   connectionString="Valid Connection String;" />  
</connectionStrings>

In the main application configuration file, you use the configSource attribute to specify the fully qualified name and location of the external file. This example refers to an external configuration file named connections.config.

<?xml version='1.0' encoding='utf-8'?>  
<configuration>  
    <connectionStrings configSource="connections.config"/>  
</configuration>

Source : https://docs.microsoft.com/en-us/dotnet/framework/data/adonet/connection-strings-and-configuration-files