I downloaded the "Magnet-O-Meter" Android app, that allows access to the device's magnetic field sensor. I see the measurement go up when I bring my phone near a magnet, which is to be expected, but I also see this happen when I bring the phone near a steel/iron object. Why is this? Does it have something to do with the magnetic field of the earth somehow being "amplified" by magnetic materials?
Because iron and most steels are ferromagnetic, i.e. capable of being magnetized. This means that external magnetic field will line up its "atomic magnets" creating an extra macroscopic magnetic field, which "amplifies" the original. Some of it remains residually even after the external field is turned off (hysteresis). Creators of the app state that it "can also be used to find static or ferromagnetic metals", and call it "Magnet-O-Meter Metal Detector". Metal detectors typically generate a magnetic field of their own and then measure magnetic fields induced by the resulting eddy currents in the metal. This works even on non-ferromagnets, but the effect is weaker, see How do metal detectors detect non-magnetic metals? The geomagnetic field is usually weak, and magnetic needles are typically pre-magnetized to make it have a perceptible effect.
It is also possible that your steel/iron objects were magnetized through coming in contact with permanent magnets or through some magnetomechanical effect, e.g. inverse magnetotstriction or surface magnetization by friction (this is often observed in drills and scissors). According to Demagnetizing Systems:
"Ferromagnetic materials are usually magnetized during various machining phases. The causes of the magnetism may be different depending on the machining: parts held with magnetic chucks; magnetoscopic checks and crack analysis; welding; bending; punching; turning; straightening; hardening induction; old forging; drilling; erosion; cutting; hardness checks; some types of laser marker; some type of induction; magnetic and electromagnetic lifters used for the movement of raw materials or finished parts; induced magnetism, caused by machining, abrasion, friction between the parts and tools."