To be honest as a drive of BMWs for many years, even driving my parents cars at 17 I've always noticed I get idiots tailgating, trying to pull away like a bat out of hell at lights, constant nudging forward.
It's not a Z4 thing but a BMW badge thing! Even if you have a model designation on the rear, you still get a tool of all ages trying to prove that their sh!t box is a million times better than your car. I'm sure when they get home they give themselves a gold star for it.
When I have my FN2 Type R I actually got respect from most performance car owners and very rarely had anyone trying to prove anything. Only twice, being a MR2 driver who hit 5th gear and I was only in 4th so I still had miles left and the other time on the A1 a bloke old enough to be my dad in a E46 330d Touring who decided as i was passing to put the foot down but it was pointless and he eventually decided to slow up and let me pass.
I get lots of drivers up my rear especially with the roof down and I've cars in front, when it's clear I'm off and their miles behind struggling to keep up. I find all white lower spec and engine 1 and 3 series drivers to be among the worst culprits. I let them get on with it, if they wish to kill themselves or blow their engine up trying to get by me, then go ahead.
Maybe if people actually drove with a bit more care and attention instead of trying to prove a point they would probably save a bit on fuel and possibly be able to afford a nicer car instead of being the jealous prick behind