what exactly does that mean? Bitstamp wont automatically withdraw funds from my account to my bank account that I've linked with them, right? So why would the money that I store on Bitstamp be sent to my bank account if I wouldn't send it there by myself ??!

Well I'm not sure if they do or even have to, I think if they were subjected to AEoI or CRS they would have to have a banking license as only banks have to report. As far as I know not even e-wallets have to report, therefore its a bit far fetched to assume that money exchanges have to. I've also never been asked for a tax number, which is a standard procedure of CRS/AEoI. As far as that goes, I would have to think they do not simply share client information but this is all guessing.

I dont think any site would share their clients information and possibly cause them trouble with tax authorities if they didnt necessarily have to.

Does anyone have fact based information?

Are they submitted to CRS / AEOI or do they exchange information with tax authorities within countries of the EU in any other form? Doesn anyone know?