Do non-mathematical fields use the appropriate level of analytic/probabilistic rigor?
I think the shortest answer is that if these other fields don't have enough rigor, the mathematicians will make up for it. In fact, a large number of important mathematical problems are just that: mathematicians working to fill in the gaps left by physicists in their theories.
On the other hand, if an economist tried to publish some grand result that used flawed mathematics, it certainly wouldn't pass through the economics community unnoticed. That being said I have read some (applied) computer science papers which spin a result to sound much grander than it is specifically by appealing to a lot of semi-relevant mathematical abstraction.
As they say in the comments, a random PhD theoretical physicist might not know measure theory, but there are certainly many mathematicians without mastery of physics working on physics equations. Similarly, an economist is unlikely to know group theory while a (quantum) physicist must. The point is that as a community we can achieve greater results.
As to the reason measure theory isn't taught to undergraduates: it's hard! Many undergraduates struggle with real analysis, and even the basic proofs underlying rigorous measure theory require mastery of a first course in real analysis, which is a stretch at a lot of universities, especially for non-mathematics majors. (Of course, at some prestigious schools undergraduate calculus is taught with Banach spaces, so I'm talking about the general undergraduate populace)
In every field more rigor is better than less, but there are costs and tradeoffs.
More mathematics means discouraging some students from the subject who could learn it and contribute. Or replacing some parts of the subject curriculum by mathematics, teaching less physics/economics/engineering/(etc). Or prolonging the training time.
Use of a complicated formalism makes it harder to communicate results. A published paper using concrete probability will be read by more people than a paper using measure theory.
For software, a formal correctness proof is great, but the time and effort involved in attaining that rigor could have been spent on securing against other failure modes, such as hacking or disrupted communication between nodes.
In engineering, simulations, prototypes, stress tests, and other forms of modeling count for more than a perfect mathematical analysis of a simpler case. Both can be trumped by "lack of rigor" in executing the blueprints (e.g. structures collapsing from the use of cheap materials, or the airplane that crashed from allowing ice on the wings). As impressive as it was for a mathematician to have detected the Intel Pentium bug, no system failure was ever attributed to that error, but a lot of engineering simulations were done using those faulty Pentia.
Classical physics describes systems that already exist and satisfy uniqueness (causality). So technical theorems about differential equations are less important than getting a model that works. Similar statements can be made about theories that are probabilistic such as quantum or statistical mechanics (which are also causal but in a different sense). Of course physicists will grab whatever looks useful from mathematics, but the test is applicability rather than formal correctness.
Theoretical economics studies toy models of complicated systems. It is a given that the models will not describe reality, and rigor allows the researchers to claim that they have accomplished something, isolating concepts from the simplified models that might be important for application to "real" economics. Many economists think that highly mathematical approaches to their field are misguided or harmful (such as creating overconfidence in models whose assumptions are rarely satisfied in practice). There have been suggestions to stop awarding economics Nobel prizes for mathematical work, or to cancel the prize because it has been promoting the mathematization of the field. The Black-Scholes-Merton formalism of continuous time arbitrage-free trading is an elegant theory that led to Nobel prizes and a spectacular hedge fund disaster (LTCM bailout), as well as a drastic expansion of the derivatives market implicated in the 2008 failure of the financial system. A lesser impression of mathematical rigor or correctness might have reduced the problems.
Because of specialization, it largely depends on what the student wants to do / know and what job they intend to get. I spent 3 years in an applied mathematics PhD program, and we were required to take a full year of graduate level measure theory and topology in the Math department. We also had our own year-long sequence in measure-theoretic probability theory, and applied functional analysis.
Statistics is one area where I feel that most departments do an inadequate job, including math departments. I work in financial research now, and there is a really widespread misunderstanding of things like p-values and the pitfalls of frequentist methods in many fields. Bayesian methods, and in particular the rigorous functional analysis behind Markov chain methods, are under-appreciated among practitioners.
However, machine learning is largely considered an interdisciplinary field involving probabilists, statisticians, electrical engineers, economists, control theorists, and even philosophers. And in that domain, to genuinely publish widely acclaimed papers, you really do have to have a deeply rigorous understanding of analysis.
On the flip side, I often find that it is the mathematicians who are most lacking, even in mathematics. I know some folks who are experts in arcane sections of group theory, or category theory, etc., and they don't understand very basic ideas about eigenvalues or the computational complexity of important algorithms. Likewise, a ton of people who are drawn to "continuous math" end up having a real lack of topology/algebra knowledge, I mean not even enough to appreciate basic important theorems like the Sylow theorems.
For the general student going to work in a technical field with some connections to research, I think it is important to get up to the point of at least one graduate level analysis/functional analysis type class -- and in particular to cover the analysis of complex functions. Aside from that, they should probably spend time focusing heavily on how mathematics has been applied to their chosen domain or sub-domain and let that guide them in choosing which upper level math subjects to really focus on.