Valve does not officially support Wine. Please contact the Wine developers instead.
https://appdb.winehq.org/objectManager.php?sClass=version&iId=19444
I'm going to close this because this issue tracker is for the native linux client.
That said, I can at least tell you that I do not experience this with wine-staging 2.12, which has an unrelated fix for https://bugs.winehq.org/show_bug.cgi?id=43315 and you may have similar success with a recent version of wine.
Your system information
Please describe your issue in as much detail as possible:
When exiting the steam client the process mentioned in the title is left behind, and it has to be killed manually. This behavior started about a month ago.
Steps for reproducing this issue: