1. Thailand like most countries, will have your primary visa record on a computer system. As far as immigration are concerned,the stamp in your identity document is secondary verification . But the computer systems are not international and so that is where the stamp in the passport is more important for the purpose of identifying where you have been and where you have a future right of entry.
2. It’s easy to get stamps transferred. It may not be the same stamp but an officers entry confirming what the system says.
3. This seems to give 3 options: noting Guiseppes comment about passport delays in Italy and your travel day of September - you may want to renew your passport now and get the stamp transferred.
4. Otherwise the thai embassy in Italy will have access and can transfer your stamp. (Edit: apparently this is not possible)
5 i otherwise again, I suspect ithe re-entry stamp in a clear part of your passport will be enough. But you could always go into immigration where you are or in bangkok and ask them to put a new stamp in your old passport..