Do all observations arise from probability distributions? Below is the quote from Karl Pearson in the book:  “The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century” by David Salsburg:

Over a hundred years ago, Karl Pearson proposed that all observations arise from probability distributions and that the purpose of science is to estimate the parameters of those distributions. Before that, the world of science believed that the universe followed laws, like Newton’s laws of motion, and that any apparent variations in what was observed were due to errors. Gradually, Pearson’s view has become the predominant one.

My question is on the use of the word observation. Does the above quote imply that any data we collect or observe in nature/physics/experiments arise from probability distribution? How about deterministic process, which surely in not probabilistic? Any expansion of the above quote for lay person would be very helpful.

Yes, would be the shortest answer. You referred to physics. Physics always disclose measurement errors or precision one way or another. The errors were always  a part of the practice of this science. What guys like Pearson did is to treat the errors as random variables. It’s nowadays common practice to follow this approach. Hence, you could say even measurements of deterministic processes are in fact sampling from distributions.
Take a look at the gravitational constant: G here, notice how with the value its uncertainty is given too. Note, this is not an inherently random quantity, it is a constant! Also read the definition of uncertainty  in NIST handbook, it is described in terms of probability distributions.
Here a snapshot from a recent physics paper: 
notice $\pm 0.03$ - the convention to report measurement uncertainty. Physicist sometimes omit it, and when they do it means that all reported digits are significant. For instance, if you see a value “127.010” it means that the uncertainty is around 0.0005, I.e. the last 0 cannot be skipped, because the authors are convinced that it is in fact zero. This is quite different from how quantities are reported in non scientific contexts, where uncertainty goes undisclosed usually
