What kind of physical processes are well modeled as poisson processes? For instance, it is easy to understand how Brownian motions appear: when an object receives many small shocks that are roughly i.i.d., the resulting path will look like Brownian motions.
What type of physical processes give rise to Poisson processes?
For instance, with one dimension thought of as time: what kind of physically processes would naturally have the "memoryless property" between spikes?

In reality, almost nothing is well modelled by the Poisson distribution
The Poisson distribution has only one parameter, so it can fit mean behaviour of a process but it cannot model the variance of a process unless that variance happens to coincide with the constrained variance of the distribution.  Consequently, in general, the Poisson model is a bad model, and anything that you might model as Poisson you are better off modelling with a two-parameter generalisation of the distribution (e.g., the negative binomial, quasi-Poisson, etc.).  Sometimes these latter distributions can be framed mathematically as mixtures of Poisson distributions, so they reflect cases where the underlying process is Poisson, but the scale of the process varies.  In any case, the advantage of the latter models is that they will tend to model the mean and variance of your process fairly well, which is really the minimum you would want from a good statistical model.
In theory, the Poisson distribution is good for modelling counts of events that occur as the limit of a binomial process when the number of trials is large and the probability of the event is small.  A standard physical example of this is when you go out in a rain-storm and catch drops of rain in a test-tube --- the number of drops that land in the tube can be regarded as an outcome from a large number of rain-drops where each individual drop has a tiny probability of landing in the tube.  Mathematically, if you have a sequence of event indicators $E_1,E_2,E_3,... \sim \text{IID Bern}(\theta)$ (e.g., that specific rain-drops land in the test-tube) then the count of the number of events in the first $n$ trials is $X = \sum_{i=1}^n E_i \sim \text{Bin}(n,\theta)$.  If you then take the limit as $n \rightarrow \infty$ and $\theta \rightarrow 0$ such that $n \theta \rightarrow \lambda$ then you get the distribution $X \sim \text{Pois}(\lambda)$.
The Poisson model is an interesting model in probability, and it gives some basic insight that we can build on with generalisations.  In reality, you almost always find that the dynamics of physical processes are more complicated than the assumptions behind this distribution, and consequently, the Poisson distribution does not generally fit well.  (Specifically, its variance constraint tends not to match the true variance in the physical process you are modelling.)  Even in the rain-drop example, variation in the intensity of the rain during the time you are collecting rain-drops is likely to lead to a situation where the number of drops caught in the test-tube is a mixure of Poissons rather than a Poisson random variable.  It would usually be modelled reasonably well as a negative binomial distribution, though other (more complex) models can be better still.
Many statistical analysts deal with this problem by using a Poisson model, but then performing an "over-dispersion test" to see if there is evidence in the data of departure from the variance implied by the model.  If the model fails this test they then switch to a generalisation such as a negative-binomial model or quasi-Poisson model.  My own view is that this two-step process is silly, and you are better off just starting with a two-parameter model and avoiding this test altogether (see this related question for discussion).
