What is the charge of linear and non-linear capacitors with same current
Of course, both capacitors will have the same current flowing through them and therefore they will end up having the same charge. Everything else would violate KCL.

The difference will be the voltage of the capacitors and the fact that the voltage ratio is not constant but dependent on the nonlinearity.

Your circuit does not contain a resistor, therefore it is problematic as the charging of the capacitors will happen instantly.