First, testing tap water and tank water for pH is different. Tank water can be tested anytime and you will get the pH at that moment in time. Tap water may contain dissolved CO2 which will add carbonic acid to the water and lower the pH; this can be temporary, depending upon the initial GH and KH and the level of dissolved CO2. As the CO2 dissipates out, the pH returns to the balance it has with GH and KH.
Second, there is a normal diurnal variation in pH in natural waters and in the aquarium. The pH will always be at its lowest in the early morning, meaning after the night darkness period and just before daylight. And at the end of the daylight it will be at its highest. Again due to the CO2. Planted tanks really show this because during daylight plants use the CO2 so the pH slowly rises, while during darkness the CO2 rebuilds and the pH slowly drops. Fish are OK with this usually, because it is gradual (over 24 hours) and minimal. Very heavily planted tanks with diffused CO2 though can be significant in this CO2 fluctuation but we are not dealing with that here. So when testing the tank water pH, if you want an accurate measure of what the pH is over a period opf time like days and weeks, always test it at the same time of day; otherwise the diurnal variation will be impacting it, which could lead you to think this or that is occurring when actually it is not.
This leads to the third aspect. As organics increase in any aquarium, the decomposition of these produces ammonia and CO2. The respiration of fish, plants and some bacteria species also produces CO2. Live plants will easily take both up, and nitrifying bacteria will take up ammonia (depending upon plant load, as plants are faster at this). The CO2 builds over time, and again this produces carbonic acid so the pH lowers. But here is where the GH and KH also factor in. The higher the GH and KH, the more they buffer the pH to prevent fluctuations. At this stage other factors also impact the pH, things like additional organics from wood, leaves, peat, etc, which also tend to lower the pH; or the opposite, calcareous substances that dissolve into the water to increase the GH, KH and proportionally the pH. All of this is natural, but it all works according to the natural laws of chemistry and biology.
So, depending upon the initial pH, and depending upon the GH and KH of the source water, the pH may or may not be affected. If the GH/KH are low, as seems to be the case here, the pH will more easily lower naturally. The effect of the organics, etc will also be stronger. But note, this is natural and gradual and will result in a very stable biological system. It is when we start to meddle that trouble enters the picture.
The obvious response to the above is to select fish that will thrive in the water. Fortunately, most soft water species also prefer acidic pH, and the reverse is also true. Some exceptions, but we needn't get into that. And letting the pH do what it will (because of all of the natural chemistry and biology which you cannot control anyway without complicated processes) is far safer.
Some aquarists think they need to start adding this and that to somehow mess with this. Wrong. This will inevitably result in a fluctuating pH and this is harmful to all fish. A stable pH even if it is not exactly in the range for the species will generally be safer. Adding so-called pH adjusters is dangerous because there is a natural chemistry in all water, and as soon as we target one aspect of that it sets off a chain reaction, and that can in many cases cause further reactions, and all sorts of problems. And fish cannot function well in any of this. And, the more of these chemicals we add, the more likely we will reach the popint when the natural buffering is incapable of functioning and the pH crashes, killing fish.
This is probably why the tap water pH is (supposedly) 7.2 and in the aquarium it will be in the 6's. The water authority might add somethingh like soda ash to increase the pH; this generally is temporary.