Short answer
Drawing these kinds of causal connections is, in principle, unproblematic. However, in single cases it is difficult to build good explanations, because it is complicated, if not impossible, to assess their validity.
Longer answer
As I understand your question, your concern is not really if Plato and Aristotle's further work is built on the prior knowledge they received in their education. Your question is rather if there is an essential determination between the professional background of Plato and Aristotle and their respective theoretical doctrines. This is a question asked in internalist approaches to sociology of knowledge (in this case: sociology of philosophy).
So, would it be safe to say that their different backgrounds had a very strong influence on their philosophies?
This claim is a strong version of the basic assumption of sociology of knowledge. Simply (and thus somewhat misleadingly) put: Externalist approaches in sociology of knowledge assume such deterministic connections in regard to social, economic, political backgrounds, internalist approaches refer mainly to the intellectual backgrounds. However, these explanations target mostly socially distinct groups of people, not single individuals. In biographical approaches to intellectual history such assumptions are often applied to single individuals, mostly, but not exclusively, to prominent figures in history.
In other words, did Aristotle tend to treat object of study as an organism because he was used to dealing with animals and plants, while Plato was more accustomed to abstract geometrical concepts?
The claim seems at first sight quite plausible ("why not?"), but this is more a problem than a feature. While the general assumption above is often taken for granted, it is difficult to assess more specific claims like yours, because they rely on very loose analogies between one domain (professional background, e.g. biological methods) and another (abstract doctrines covering a lot of other things outside of biology). This is a problem, because with hindsight many other connections (including other analogies) can be set up at will, without any good methods to test them.
This problem is known as the construction of just-so stories, especially when these "stories" are used as explanations ("Aristotle tend to treat object of study as an organism because he was used to dealing with animals and plants").
To sum up: Drawing these kinds of causal connections is, in principle, unproblematic. However, in single cases it is difficult to build good explanations, because it is complicated, if not impossible, to assess their validity.
If you are interested in this kind of explanations in the history of philosophy, there is an ambitious book which I can recommend: Randall Collins, The sociology of philosophies: a global theory of intellectual change, HUP 2000
Here's the blurb:
Through network diagrams and sustained narrative, Randall Collins traces the development of philosophical thought in China, Japan, India, ancient Greece, the medieval Islamic and Jewish world, medieval Christendom, and modern Europe. What emerges from this history is a general theory of intellectual life, one that avoids both the reduction of ideas to the influences of society at large and the purely contingent local construction of meanings. Instead, Collins focuses on the social locations where sophisticated ideas are formed: the patterns of intellectual networks and their inner divisions and conflicts. According to his theory, when the material bases of intellectual life shift with the rise and fall of religions, educational systems, and publishing markets, opportunities open for some networks to expand while others shrink and close down. It locates individuals -- among them celebrated thinkers like Socrates, Aristotle, Chu Hsi, Shankara, Wirt Henstein, and Heidegger -- within these networks and explains the emotional and symbolic processes that, by forming coalitions within the mind, ultimately bring about original and historically successful ideas.
