Same problem here, “Charging” mode won’t charge it connected to my computer usb-plug. All other modes work for charging, even not selecting one at all (second tap deselects the current mode so that none is arrowed; @tkos maybe trying this could help with your car problem?) Turned off the phone seems to be charging, too. Using Fairphone Open 16.10 and Lenovo Z360 Laptop with Ubuntu 16.04LTS. So this problem seems not to be one of the computer operating system (as it happens with different systems), but a software/driver problem of the Fairphone.
I also have problems with my chargers (HTC 1A and Samsung 1A with different cables), although those are a little different. I’m not sure if there is a connection, but it would be interesting to hear other experiences on this. Charger problems are mainly discussed here: Charging the FP2 / cable problems