I suppose there is a certain sense in which this is true, at least concerning The Scientific Method presented in e.g., science education classes and textbooks (up to and including college), popular science books, popular science magazines and online equivalents, etc. As we don’t actually use The Scientific Method in scientific research and never have, and as it exists only as a kind of pedagogical lie and popular myth, it would be rather difficult for any practicing scientist to have any assumptions about The Scientific Method that is relevant to their work.
But if you mean refer to “scientific methods” actually used by practicing scientists, then there are a whole slew of assumptions that vary widely from field to field but are always present (explicitly or implicitly). Then there are the sets of assumptions practicing scientists rely on (again, both explicitly and implicitly) that vary within any given field based on different approaches and often correspondingly based on (usually tacit, even unconscious) metaphysical, philosophical, even political worldviews.
That there are always assumptions at play in scientific research is perhaps less important for non-scientists to understand (at least initially) than is the fact that The Scientific Method that most of us learned in school (and which may have been reinforced elsewhere, such as e.g., popular science books, articles, podcasts, etc.) is a myth. Interestingly, its form (at least in the US) has a rather singular source, being largely borrowed with little variation by the publishers of the early 20th century (seeking to meet a new demand from the rapidly growing education system) from an influential book on the relevance of logic by John Dewey:
How We Think
"Dewey was not the first to try to separate the intellectual process of scientific reasoning from the laboratory method of instruction to which it had been wed since the 1880s...
Ironically, none of his discussions of science education clearly laid out what became known as the steps of the scientific method. The work that spelled these out and that was ultimately responsible for reifying the five-step process in the nation’s classrooms was How We Think, a short textbook for teachers that Dewey described as “an adaptation of a pragmatic logic to educational method…
Despite his borrowing from the sciences, it is important to understand that Dewey did not try to provide a stepwise account of how scientists went about their work. He aimed rather to describe reflective thought in the most general sense-to detail the way people used thinking as an effective guide to practical action.”
Rudolph, J. L. (2005). Epistemology for the masses: The origins of “the scientific method” in American schools. History of Education Quarterly, 45(3), 341-376.
The more global history (and even the American history) of this idea of The “Scientific Method” is more nuanced, of course. It is so pervasive that decades of attempts by scientific organizations such as the AAAS or NAS, a vast number of conferences, countless papers published in journals on education or more specifically on science education, and more, have all failed rather spectacularly in their attempt to reform the manner of science education to better reflect actual scientific practice and dispel the myth of The Scientific Method.
There is in general no clear cut distinction between the methods used for the growth of scientific knowledge by (among other things) scientists engaging in scientific research on the one hand, and methods used in other fields and areas of inquiry on the other. For example, what makes certain developments in mathematics “science” (usually physics) and others “mathematics” is often context. An unfortunate illustration of this and how it can slow progress is the development of gauge theory. In the physics community, this misnomer was initially proposed by Weyl and then largely (and almost immediately) rejected. It was later rediscovered during the crisis after the short-lived triumphs of QED and is now at the foundation of the standard model after the groundbreaking work by physicists (e.g., C. N. “Frank” Yang and Robert Mills of Yang-Mills theory fame).
Meanwhile, mathematicians practically next door to the leaders in the development of gauge theory were working on much the same kinds of problems as a continuation of more general mathematical developments in manifolds, differential geometry, etc.
So while physicists were developing the mathematics of gauge field theory, mathematicians were independently doing much the same work (at much the same time, albeit a bit earlier) in terms of fiber bundles, sections, connections, etc.
About a century earlier, such distinctions between physical theory and mathematical research would hardly have been possible, and indeed the historical development of field theory is a marvelous example of empirical findings providing motivation for theoretical development which was in turn the motivation for and then beneficiary of advances in mathematical physics, all by scientists for whom these distinctions are largely anachronistic and likely bordering on nonsensical.
Other examples include areas such as history, anthropology, and linguistics, in which research in the same field using different methods with different aims may fall under a different scientific discipline (e.g., biological anthropology vs. the more qualitative, social science anthropology or cognitive linguistics vs. historical/comparative linguistics) or fall outside the sciences altogether.
