If I acquire, for example, an Amulet of Life protection when I'm level 2 when its modifier is +1, does its modifier increase to +2 automatically when I reach level 7, or is that counted as a completely different item which I then have to "refind"? Is the answer the same for all other items and weapons in the game?
No. If you have a Amulet of Protection +1, it stays an Amulet of Protection +1. You will have to find, buy, or make an Amulet of Protection +2 if you want that instead.
All items work this way. Although there are artifacts (such as the Staff of Fraz-Urb'luu) that can increase their enhancement based on other parameters, they are the exception rather than the norm.
It is a completely different item that you can upgrade with the appropriate magic ritual (Enchant Magic Item). This happens to all items in the game, maybe except some artifacts that become more powerful as their owner grows.