If you go into 'Network and Sharing Center' then, 'Change adapter settings' I think it shows you your connections there.
If so, what I would do for neatness in the 'Network and Sharing Center' would be the following;
On Computer A, rename the connection called 'Network' to 'ASUS' by right clicking on that network connection's shortcut icon and select rename from the menu that appears.
On Computer B, I would left click on the 'Unidentified network' connection then choose the 'Organise' drop down menu on the toolbar and then select delete.
It's just a computer being a dumb computer really, it does the same thing with wireless connections, just because you connect to a wireless connection with the same SSID it doesn't know that you are actually connecting to the same router that you have done previously as SSID is just a variable that many people could choose as the same thing.
If you can rename them then it just makes it a bit simpler if you end up with more connections at any point, that the screen name you've given a connection is how MS probably refers to it in some dialog box.