I think the starting point in answering that question is to distinguish two separate offences. In proposed section 56.1, that's the offence in relation to government-issued identity documents. Identity theft doesn't require reliance on a document. That offence can involve mere possession of the information, even if it's not captured in a government-issued document.
If an immigrant comes and doesn't have papers, they still have an identity--an address, a name, a birth date, and other identifiable information such as financial account numbers. If someone obtains any of that information with the intent of using it to commit an offence, even in the absence of there being legitimate government documents, that can still be captured as an offence. But if a person has no documents, then there are no documents in respect of which another person can commit an offence. In other words, you simply can't commit an offence in respect of documents that don't exist. That is for the offence under proposed section 56.1 in relation to government-issued documents.
Whether or not or how that person might obtain government documents would not really be a matter for criminal law. That would be a matter for Citizenship and Immigration authorities, or provincial authorities. But you can't commit an offence in respect of something that doesn't exist. An individual like that would be protected in terms of the misuse of their information, but if there are no documents, proposed section 56.1 wouldn't apply.