Learning in the car you will actually drive has obvious appeal. You know where the controls are, the mirrors are already set for you, and the vehicle is not an unfamiliar variable on top of everything else you are trying to manage. Many instructors will still decline, and their reasons are worth understanding before you push.
The Dual Controls Problem
This is the main obstacle. A teaching vehicle has a second brake pedal, and frequently a second clutch, letting an instructor intervene in a fraction of a second. In your car they can only tell you to brake, which takes far longer to reach your foot than their own foot reaches a pedal.
For early lessons on public roads, that difference is significant. Some instructors will teach in a learner's vehicle only once basic control is established.
Insurance Is the Other Blocker
Your policy must cover a permitted driver, and it must cover a paid professional instructing from the passenger seat. The second condition is the one that catches people out, since many policies that happily cover a learner with a supervising relative exclude commercial instruction.
Call your insurer and ask specifically. Get the answer in writing. An uninsured lesson is a serious problem for both of you, and the instructor's own policy will typically not extend to a vehicle they do not control.
Condition and Equipment
An instructor teaching in your car will want it roadworthy without argument. Brakes, tires, lights, signals, wipers, and mirrors all functioning. Registration and inspection current.
Many will also want an additional interior mirror fitted so they can see behind without turning, and some require a visible sign indicating a learner is driving.
What You Gain
Genuine advantages, particularly later in the process. Familiarity with your own vehicle's dimensions, biting point, and blind spots is exactly what you need before driving alone, and a learner who trains entirely in an instructor's car sometimes finds the first solo drive in a different vehicle unsettling.
Road Test Considerations
If you plan to test in your own car, confirm it meets your state's requirements well in advance. Working seatbelts, functioning brake lights, valid registration and insurance, and no warning lights illuminated. Vehicles are turned away at test appointments regularly.
Check the vehicle requirements in your state, and see whether driving schools rent cars for the test if yours does not qualify.