According to the set of technical regulations written by the FIA, to be considered "legal" by F1 standards, the cars must follow a set very specific and detailed rules that affect each part of the car, including the following:
- aerodynamic Components
- power unit
- fuel system
- coolant systems
- charge air cooling
- electrical systems
- transmission systems
- steering systems
- brake system
- car construction
- survival cell (chassis)
- safety structures
- engine oil
If any of these rules are broken for any part listed above, then the car is not considered as a legal F1 car.
For example, the following article states that the plank assembly below the car must be fitted, so a car without a plank would not be deemed an F1 car.
3.5.9 Plank Assembly Below the central surfaces of the Floor Body, the plank assembly must be fitted. consisting of the plank, the skids, and
the mountings. The requirements of this article must be satisfied when
considering both sides of the car.
There are obviously many more parts that are made "mandatory" through these sets of regulations, such as for the DRS. But I noticed many parts mention that a part declared as "X" must follow a set of rules, such as below, but it doesn't say that the part must be on the car. So perhaps reading through the whole regulations would be enough to determine which parts are mandatory and which aren't.
3.10.2 Pylons Bodywork declared as “Rear Wing Pylon” must: a. Lie within RV-RW-PYLON. 2022 Formula 1 Technical Regulations 31 15 October
2021 © 2021 Fédération Internationale de l’Automobile Issue 7 b. On
any Z plane, when considering both sides of the car, only contain no
more than two closed symmetrical sections with a maximum total area of
5000mm2 . This does not apply if the complete section lies within 30mm
of the Exhaust Tailpipe defined in Article 3.8.2. For any complete
closed section outside of RV-TAIL-EXH, the thickness of each section
may not exceed 25mm when measured in the Y direction.
The full set of rules can be found on the FIA's website under regulations: FIA Regulations. The set of regulations concerning the legality of F1 cards can be found under "technical regulations".