It can happen by chance or convergent evolution. Mutation rates are sometimes artificially raised to increase the probability of it happening. https://en.wikipedia.org/wiki/Mutation_breeding
It's extremely hard to selectively breed fish DNA into a plant. Convergent evolution doesn't end up with the same DNA sequence. My favorite example is the protease catalytic triad, where 20 or so different evolutionary routes have converged to the same chemical function, even though the DNA for those different superfamilies are different.
The odds of mutation breeding producing a new gene (or more likely, changing an existing related gene to correct form) is incredibly small, and the result would be very expensive. Each generation would need to be sequenced, and the ones whose DNA is closer to the target re-bred. You want the mutation rates to be low enough that the rest of the plant isn't killed, so that's what, a mutation somewhere in the target gene every 1,000 generations? Assuming a short sequence of 100 base pairs, and perfect mutation gives 100,000 generations. Assuming a life cycle of 10 days means this will take over 2,500 years.
I deliberately low-balled these numbers.
So yes, "by chance" it's possible, in about the same way that a universe of monkeys could type out Hamlet.
How do these observations lead to a breeding technique where a plant ends up with fish DNA?
You'll notice the the events listed are either in related species, or from ecologically very tightly coupled species (eg, host and gut bacteria), and seem to imply time scales and population counts which are much higher than would be economically feasible for breeding.
Because there's no such thing as 'fish DNA'. There's just DNA. Its a code, and code doesn't 'belong' to any organism or know where its at. So transfer of DNA from one organism to another, like monkeys typing, increases the rate of change in species well above that of mutation alone.
