Grouping SQL Server Connections By Connection Pool
You can get a little closer if you add host process id but I don't think you'll be able to get an actual connection pool count 100% from server data because many connection string attributes are processed on the client and never sent to the server. This might be close enough for your needs, though.

SELECT
      s.login_name
    , s.host_name
    , s.program_name
    , s.host_process_id
    , COUNT(*) AS connection_count
FROM sys.dm_exec_sessions AS s
GROUP BY
      s.login_name
    , s.host_name
    , s.program_name
    , s.host_process_id;


A client-side exception like "timeout period elapsed prior to obtaining a connection from the pool" should be raised when the pool size is reached an another connection cannot be acquired within the connection timeout limit.