The responsibility of the biological father and mother of the child does not change because the parents get divorced. Their responsibilities towards the child remain the same.
The stepfather does not have explicit financial responsibility towards the child unless it is stipulated in the marriage contract that he does so. However, the child does become mahram to the stepfather and the stepfather is expected to act towards the stepchild with every degree of kindness and affection that is customarily expected.
Indeed, if it is customarily expected in the society in which he lives live for the stepfather to provide financially for the stepchildren – or to share with the biological father that responsibility – then it is incumbent upon him to do so.
This is based on the principle of Islamic contract law that: "What is known by custom is the same as what is explicitly stated as a condition." Ibn al-Qayyim discusses this legal principle at length in I`lâm al-Muwaqqi`în, where he clarifies the case of the Prophet (peace be upon him) prohibiting `Alî from taking a second wife while being married to his daughter.
Therefore, in a culture where the stepfather is expected to assume financial responsibility over his stepchildren – which is the case in most Muslim and non-Muslim societies around the world – the responsibility is understood to be implicit in the contract of marriage, and therefore it is binding even if it is not explicitly stated. Therefore, in most countries, it will be the case that the stepfather will have financial responsibilities towards his stepchildren. This will be implicitly understood as being one of the conditions of the marriage contract.
The only way that the stepfather in this case will be exempted from the spending what is customarily expected from him to spend is if it is explicitly stated in the marriage contract that he is not financially responsible for his stepchildren. This condition of exemption will be valid, since the stepfather's liability to spend on the children is based on a customary condition and is not an inherent and unalterable pillar of the Islamic marriage contract.
And Allah knows best.