Amplify signal going to motor of kids electric car
Since the motor can take 12V, you can try to use 12V.
6V+6V in series connection to get 12V. Or just buy a 12V battery.

Edit for misunderstanding the question:
For Electronic 6V input:

Use buck converter or DC-DV converter 12 to 6V use separate cable for connecting to your electronic device.
Or you can try to draw 6V from your one of your 6V battery (6+6 in series) and the 12V to your motor
This is why I post the picture of 12V in series.

For motor:

Did it really can handle 12V or there is another driver for the motor (BLDC driver)?
If it is a direct DC motor (without any controller) then you can connect it to your motor via the relay, the relay should be 6V which is I guess from your original battery this is all operated with 6V. With this relay, you connecting it to your battery for your motor.

I hope this one clears it up. It should be simple.
Edit 2 for someone asking the source or link of the picture above.
CLICK HERE
Edit 3:
Here is the diagram for easy to modify your electric car kids.

The red line is positive(+) and the negative(-) is black
Because the original question doesn't provide the board picture and connection, I can only assume the board has 2 cable inputs for power (+ and - from 6V) and the output is 2 cables to the motor (6V output?)
Please check whether your toys have manual gear for reverse or it has digital switching on your board to makes polarity changing. If your toys have manual gear for reverse or it doesn't even have a reverse function then my diagram could work
If you are worried about the unbalanced draw on each battery, you can use an extra device such as a converter or buck converter from 12V to 6V DC-DC before your electronic board. I hope this helps you.