Right I've just been to my car to check if my Z2 plays automatically when it connects to my car's bluetooth and it doesn't, I have to press play on my car stereo or the walkman app for music to play.
I have to ask though, with your original Xperia Z, what version of Android is it running? I ask this because it's only been since Android 4.3 that bluetooth audio and associated protocols have been fully put to use (excluding any 3rd party bluetooth stacks). My One X is on android 4.1 and only responds to previous and forward from the car stereo the same was true with my HTC Desire on Android 2.3. Z2 on 4.4.2 responds to all the controls and also sends song information to the car stereo, the car stereo is a fairly old JVC KD-BT11.
I think it's more likely that your car stereo is trying to play music automatically when it connects and may have tried to do it with the Z (if the Z wasn't on 4.3). This is all just mostly guesswork, otherwise I have no idea why it's happening.
Phone History: -
Ericsson A1018s > Ericsson T20e > Nokia 3510i > Motorola V600 > Motorola V3x > Sony Ericsson K800i > Sony Ericsson Xperia X1 > HTC Desire > HTC One X > Sony Xperia Z2 (Current)