Nope... we talk about specific countries... for some specific reasons it can make sense to have two valid passports. And yes... it's possible, even issued from the same country.
You can have as many different visa (from different countries) as you want. Only for some specific countries it might be a good idea to use a second passport... just in cases... ;-)
Anonymer Teilnehmer The Immigration has your visa in their database, bonded to your passpaort. If someone can tell what to do in this case, then the Immigration, no? Somehow you need to inform them about your new passport and they will change something... they might even give you a kind of "transfer stamp" in the new passport... not sure about that.
If the law (or the rules) in a country require that you have an account in that country (Non-Imm-Visa), than the same country cannot have a law (or rules) that excludes you from open an account. That's absolutely ridiculous...