The difference between banks and other financial institutions such as web wallets (even if they have the word bank in their name) is that banks can loan money to borrowers while web wallets can't. This is a distinguishing feature of any genuine bank (i.e. attracting money from depositors and loaning it to borrowers).
It was a soft comparison, and not a direct hit. Nothing prevents web wallets or exchanges to hand out bank-like features such as loans, or exchanges that allow their traders to loan out their coins through the exchange's platform (which is already a reality). The main fact is that people still take the term "Bitcoin bank" way too serious. They see it as an official institution which it's obviously not.
Regarding governments never supporting something which could potentially bring harm to their economy, it is also debatable. Bitcoin would most certainly cause damage to a fiat monetary system as such, but not real economy.
If the fiat monetary system is being hammered on, how would that not translate into the real world economy experiencing the negative effects as result of that? They are closely tied to each other.
In fact, anything that would facilitate the exchange of goods better than the currently existing system could only push the real economy forward, not hurt it
Correct, but that's not how governments look at it. Especially when it comes to something as Bitcoin that they have zero control over. In that regard, they are better off creating their own digital currency with the exact features of Bitcoin, but with the government as central point.