Here's a fun fact. Nothing you measure is what you intend to measure. The corrections of these measurements to satisfy the objectives form a good chunk of development flight test by the airplane manufacturer. Even after the airplane type is certified and enters into service, the corrections may continue!
Now specifically for AOA ($\alpha$) and AOS ($\beta$). In development flight test, an accurate measurement of AOA and AOS is typically done with a nose-boom that places it outside of the aircraft boundary layer (as reasonably can). But there's an error even with that, due to Mach, manufacturing tolerance, etc. Flight path reconstruction could be performed to further decrease the errors, if needed. Usually, the boom AOA and AOS measurements are pretty close to the actual AOA defined relative to the body axis and used in wind tunnel.
In service, AOA is measured with vanes located at the nose of the aircraft. Even at low speed where Mach effect is small, these measurements are not the aircraft AOA. They are local AOA due to upwash and boundary layer. For non FBW aircraft, AOA measurement is used primarily for stall warning and stall protection (SPS). There are lookup tables in the SPS that register the stick shaker and pusher (if required) angles, as a function of the local AOAs. An extra mapping, as a function of Mach, configuration, whatever, is required to transform the local AOAs to aircraft AOA, should the design need it.
Development flight test airspeed is usually measured by the nose-boom, often augmented by an additional trailing cone. The trailing cone is extended behind the aircraft to measure the far field static pressure. Total pressure is measured by the boom. The resulting impact pressure needs to be transformed to dynamic pressure to output airspeed.
In service airspeed is measured most often by pitot-static tubes at the nose. Due to boundary layer, static source error correction (SSEC) would be required. These lookup tables are stored in the air data system (ADS). There may also be a mixing of the left/right static ports to decrease the effect of asymmetry.