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.

Probability is not fundamentally about the nature of the world (which may, or may not, be deterministic) but about what you know about it.
Consider this example.

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*You are sitting with your friends, Alice and Bob.


*I have a standard deck of cards, and shuffle them well.


*What is the probability that the top card is the Ace of spades? Clearly $\frac{1}{52}$.


*I show the top card to Alice, but not to you or Bob.


*If I ask Alice what is the probability that the top card is the Ace of Spades she will surely answer either $1$ or $0$, but not $\frac{1}{52}$.


*But If I ask Bob, he will still have to say $\frac{1}{52}$.
The point is to demonstrate that probability is not fundamentally about reality, but about your knowledge of reality. The cards have not changed their order.
Consider tossing a fair coin. What is the probability that it will land heads? $\frac{1}{2}$?
But in fact tossing a coin is a deterministic process, at least by the standards of modern physics. Scientists have built machines which can predict a coin toss from video of the first milliseconds of motion. Some magicians have trained themselves to toss coins so accurately that they can get either heads or tails at will. I suppose they do this by finely calibrating the force with which the coin is flipped, and the moment at which they catch it, so that they know exactly how many times it has turned over.
But if I, not having trained to do this, were to toss a coin, I couldn't predict the outcome. When I flip a coin it might sometimes go twice or three times as high, or have twice or three times as much angular momentum, as at other times. At best I might say it had turned over between 3 and 15 times. So it is clear that for me, even if I took notice of which way up it was to begin with, my probability will still be close to $\frac{1}{2}$.
Again, the point is not that the process is not deterministic - clearly some scientists and magicians can do it - but that I don't know the parameters of the deterministic function to a high degree of accuracy. My initial ignorance, or imprecision in knowledge, integrated over time, expands out to cover the entire space of possible outcomes, such that I have no idea which way the coin will end up.
Back to your question

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?

Go back to the coin toss. The coin toss is in modern terms an entirely deterministic process - if we start with the initial conditions and integrate over time we will get the answer. What makes it "random" is we do not know the initial conditions with enough accuracy to predict if it will be heads or tails. We can take our estimates of the initial conditions and their error bars, and run millions of deterministic Monte-Carlo simulations using slightly different initial conditions, and in each individual simulation there will be an answer, but the answers will be different, and the ratio of heads to tails will be about $\frac{1}{2}$.
So another way of thinking about it is to say that, supposing for a moment that the universe is deterministic, then the "probability distribution" is the weighted distribution of the time integrals of all possible pasts. That is, every possible past - those which we don't know to be false - integrated deterministically through time to the present.
(Kolmogorov will be turning in his grave no doubt.)
So in this view, an observation is a ground truth, and we can integrate backwards through time to eliminate possible pasts which would not give rise to that observation.

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*If you have just been dealt the Ace of Hearts, that means I wasn't dealt it earlier.

In summary,

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*I would not say that an observation arises from a probability distribution.

*There is a reality out there which gives rise to observations.

*Observations give us information about that reality (specifically about the past) and we can combine observations to create a model of the past which allows us to make predictions about the future.

*That model of the past is the probability distribution referred to.

