Connecting two similar, equal voltage batteries in parallel
Can I put these two in parallel? Some sources say that the voltage has to be the same other sources say everything must be exactly the same.

Two batteries with the same nominal voltage rating, can easily have different open circuit voltages. When two batteries with different open circuit voltages are connected together in parallel, current will flow from the battery with higher voltage to the battery with lower voltage, until the batteries are equalized.
Hopefully, the total charge transferred from one battery to the other will be small. However, if the batteries have different chemistry, or even different geometry, the charge transferred could be significant. If both batteries are fully charged, and they have different characteristics, it is possible that one battery might become overcharged. That would be bad.
You are safest if the batteries are identical. Identical voltage, identical chemistry, identical geometry, identical manufacturers. If they are not, you take some risk. How much risk? That is impossible for me to say.

can I also use the built-in charger to charge both at the same time or do I have to modify it somehow?

Without knowing the details of the charger, it is impossible to know whether it will charge the batteries correctly, whether it will damage the batteries, whether it will leave them less than fully charged, or even possibly damage the charger, although I think that last possibility is rather unlikely.
Edit:
@StainlessSteelRat has provided a useful point of reference regarding the risk associated with using batteries in parallel, and what is considered good practice within his field:

I deal with the marine industry, which uses battery boxes with parallel batteries. A bad battery and we switch out all batteries in box for same batch.

I would also like to concur with his evaluations

Nothing of OP info makes this a good idea. Including a new battery vs used battery.


Different manufactrers, chemistries, ages, electrolytes, capacity = bad idea.


And if everything is exactly the same? Still not a great idea? Id like to stay on the safe side considering this is a child car.