As of 2019, which discrete nonlinear, time-invariant systems with memory are considered "easy" to model and identify?
It is very strange phenomena that one object is completely dropped out of attention of researchers. It is Urysohn operator. First of all Urysohn is equivalent to multiple parallel Hammersteins and Urysohn followed by static nonlinearity is a model of any deterministic dynamic object, it maps any given input to any provided output. I obtained Ph.D. in modelling dynamic systems as Urysohn operators in 1991. Since that time I showed how to do that to many people, I published articles, I maintain the site, explaining it, and for this almost 20 years noone started using it, there is no single reference on my articles. People write 10,000 lines to identify Hammerstein, I show how to identify Urysohn by 10 lines of code, write here. Look 

for (int i = T - 1; i &lt; N; ++i) {
   double predicted = 0.0;
   for (int j = 0; j &lt; T; ++j) {
      predicted += U[(int)((x[i-j] - xmin) / deltaX), j];
   }
   double error = (y[i] - predicted) / T * learning_rate;
   for (int j = 0; j &lt; T; ++j) {
      U[(int)((x[i-j] - xmin) / deltaX), j] += error;
   }
}


That is it. It replace 10,000 line Hammerstein identification. It published, it showed, why noone use it I don't know. Google show my articles on "Urysohn identification" at the top. People don't search. 

Explanation to code. x[i], y[i] elements of input output arrays. U[k,m] elements of model matrix. T is one size of matrix U, another one is determined by deltaX, discretization interval, learning_rate is similar to LMS.