from your attached PNG, it appears you're using SQLExpress. By Default this doesn't start on TCP port 1433. Check the ports in the Configuration Manager->Network Config->ProtocolsFor...-> right-click TCP/IP, choose properties and click IP Addresses tab
Scroll to the bottom to set TCP Port under IpAll section to 1433, restart SQL Server, then try again (or use whatever ports appear to be defined here in your connection string)
I'd also recommend using SQL Server users for remote clients rather than windows - this way it's nicely separated from windows and domains etc.
Make sure your SQL Server is configured for mixed mode:
https://docs.microsoft.com/en-us/sql/database-engine/configure-windows/change-server-authentication-mode?view=sql-server-2017
you can then create a LOGIN (for the database server) and a USER (for the specific database you want them to access) via SSMS, or you could use this script:
-- Creates the login remoteuser with password 'Pa$$word11'.
CREATE LOGIN remoteuser
WITH PASSWORD = 'Pa$$word11';
GO
-- Creates a database user for the login created above.
CREATE USER remoteuser FOR LOGIN remoteuser;
GO
--grant remoteuser 'datareader' rights
EXEC sp_addrolemember N'db_datareader', N'remoteuser';
--grant remoteuser 'datawriter' rights
EXEC sp_addrolemember N'db_datawriter', N'remoteuser';
You can do all the above through SSMS - add a login under the Security node, then in properties for the login set up database-level permissions too. The script is a bit more managable if you ever want to repeat the process though.
Connection string is then something like :
Data Source=192.168.0.55\SQLExpress;Database=MyDatabase;user=remoteuser;password=Pa$$word11