Bluetooth Headset not recognizing my phone as no phone
It appears to be a pair of headphone, not headset. In other word, it doesn't have a microphone builtin.

Hence the product probably isn't supposed to be used for communication, as in, it doesn't support HFP/HSP in Bluetooth but only A2DP.

As far as I know, Android does not have audio route written for outputting "voice" to A2DP-only device.

EDIT: You may also see if the settings of the paired item has a Call/Phone Audio checkbox in addition to a Media Audio one. If it does and the box is "checkable" (i.e. not greyed out), then you should be able to use it for communication by "checking" the box. After that (No Phone) should be gone.