Very weird bug

My League client is having the weirdest bug I have ever seen. There are 2 computers with League installed in my house. One of them is a windows PC and the other is a Mac. When I log into my account using the Mac client, the windows client automatically stops communicating with the server and crashes. It's not a coincidence because I've tested it several times. I had no idea that one client could mess with another one in a different computer. If only the windows client is open, then it works fine. What could it be?