Congress and the president control processes such as passport issuance by passing laws, which are subject to judicial review. Within the framework of the three-way checks and balances, therefore, the federal government as a whole is responsible for determining the rules for such a process.
The income tax process is also defined by federal laws. Since the federal government controls both processes, it is free to create interdependencies between those processes. For example, it can require passport applicants to give their social security numbers if it wants to be able to cancel the passports of those who are delinquent in paying their taxes.
The "bill by congress" does not become binding law until signed by the president, as you no doubt are aware. Independent government agencies may create requirements for inter-agency business by agreement, as long as that does not conflict with the law. Some statutes will say things like "... or as so-and-so may by regulation provide." This hints at the existence of another important code, the Code of Federal Regulations (CFR). These are mostly (or perhaps entirely) regulations promulgated by the executive branch, pursuant to authority granted by the constitution or by statute.
For what reason could a US citizen possibly be denied a Social Security Number? The only cases I'm aware of where someone is ineligible is the case of a foreigner who is not allowed to work in the US. Such a person is also, of course, ineligible for a US passport, so there is no conflict in the requirement to disclose the SSN in the passport application.