Let me start off by saying that there is no doctrinal unity across all Catholics in any other than a tautological sense (as in: "there is authoritative doctrine, and whoever doesn't adhere to all of it is not a Catholic"). I know people who are Catholics who find the part of the Nicene creed that talks about the "apostolic Church" ridiculous, and there are numerous Catholics who find the idea of a virgin birth ludicrous. I've even seen bishops skirt the question of virgin birth, which doesn't mean they don't believe it but it does indicate an unwillingness to publicly commit to what everyone can easily determine to be Church doctrine. You will be hard pressed to find two Catholics that agree on absolutely everything as well, but certainly to a far lesser degree than with Protestants, and the disagreement is likely to be on less central issues.
In (Sunni - I'm not that familiar with Shia) Islam, there is no formal hierarchy of authority on doctrinal matters, i.e. there is no clergy, and certainly no person who can declare a dogma after the death of Muhammad. However there is also not the doctrinal chaos that Protestants generate. What there is are scholars of the religion (ulama) who interpret the foundational texts according to different methodologies; with regard to fiqh (jurisprudence), four schools of thought (each with its own methodology) have been dominant historically almost to the exclusion of other schools, and with respect to aqida (belief), there have been three schools (Asharites, Maturidites, Atharites); the schools of belief differ in minor issues.
What ensures a certain degree of unity in doctrine is the scholarly discourse that weeds out lines of argument that are considered methodologically unsound - scholars do not take the texts that they find, read them however they want, and declare what they come out with to be valid opinions; they have to argue their opinion, and the discourse is committed to weeding out opinions that are not justified by a recognized methodology.
On central matters, the discourse arrives at a consensus (ijma), and those central matters are somewhat few. The eminent medieval scholar al-Ghazali (who I refer to here because on this issue, he gives a minimal outline that every Muslim scholar will agree on without hesitation) names three central tenets:
- belief in Allah (the one and only God of Abraham and the other prophets)
- belief that everything Muhammad said was true (which includes that the whole Quran, letter for letter, is the literal word of Allah) and that he did not make any mistakes in matters of teaching the religion
- belief in Judgment Day
These three already imply the parameters of what is and what is not necessary belief to some precision. Whatever is reliably demonstrated to have been stated by Muhammad, with the wording unambiguous enough to be interpreted in more than one way, relays necessary beliefs. When the transmission of a tradition of what Muhammad said (hadith) is not completely without doubt, then such a tradition does not by itself imply a necessary belief; however, the scholars being unanimous about a doctrinal point, this engenders necessity according to many scholarly opinions.
As with the Catholic Church, the methodologies Islamic scholars rely on are not entirely axiomatized, but in a sense they are more stringent than Church dogma: declaring a dogma in Catholicism is an act of justification by authority instead of justification by reasoned argument, while in Sunni Islam ijma is not arrived at by anything other than justification via accepted methodology. In this sense, Islamic beliefs on at least central and major issues flow from a somewhat rigorous discourse in the sense that they have to be justified from (textual) evidence (Quran and hadith) via a collective process where peers evaluate and critique each other according to a set of methodological tools that are thought to be mostly reliable. The major split Sunni-Shia comes from different textual evidence being admissible in the discourse (because Sunnis trust other sources than Shias).
This already indicates how one becomes an Islamic scholar: you study the texts, you learn the methodology and the necessary tools (which includes the language, its grammar, and its idioms and styles), and you study the arguments used in the discourse on the issues you study. Whether you have learned enough is a matter of judgment of established scholars. Your teachers on specific subjects traditionally give you a formal declaration that you are accomplished enough in a given field/discipline so you can teach it yourself - this declaration is called ijazah. As for your opinion finding authority overall, that is a matter of discourse comparable to that in secular jurisprudence - if other scholars recognize you as putting forward valid opinions, that makes you a scholar (alim).
The lay Muslim is expected to learn about beliefs as they are considered necessary by scholars, and disagreeing with all scholarly opinions on a given question of belief while knowing that the scholars reject the opinion you embrace is at the very least sinful, and - depending on the importance of the issue, the discrepancy between the belief the person embraces and what the texts say, and the reliability attributed to the texts - liable to constitute disbelief.