I cannot judge right now the merits or demerits of the specific book you mention. Joonas Ilmavirta addresses in their answer whether your question is appropriate and contains some remarks about the book's claims. The term nonlinear algebra has been used by others though. See, for instance, the draft text by Michalek & Sturmfels, the research group at MPI Leipzig, and Sturmfels' lecture series.
Linear algebra is the foundation of much of mathematics, particularly
in applied mathematics. Numerical linear algebra is the basis of scientific
computing, and its importance for the sciences and engineering can hardly
be overestimated. The ubiquity of linear algebra masks the fairly recent
growth of nonlinear models across the mathematical sciences. There has
been a proliferation of methods based on systems of multivariate polynomial
equations and inequalities. This is fueled by recent theoretical advances, efficient software, and an increased awareness of these tools. At the heart
of this lies algebraic geometry, but there are links to many other branches,
such as combinatorics, algebraic topology, commutative algebra, convex and
discrete geometry, tensors and multilinear algebra, number theory, representation theory, and symbolic and numerical computation. Application areas
include optimization, statistics, complexity theory, among many others.
Nonlinear algebra is not simply a rebranding of algebraic geometry. It
is a recognition that a focus on computation and applications, and the theoretical needs that this requires, results in a body of inquiry that is complementary to the existing curriculum. The term nonlinear algebra is intended
to capture these trends, and to be more friendly to applied scientists. A special research semester with that title, held in the fall of 2018 at the Institute
for Computational and Experimental Research in Mathematics (ICERM) in
Providence, Rhode Island, explored the theoretical and computational challenges that have arisen, and it charted the course for the future. This book
supports this effort by offering a warm welcome to nonlinear algebra.
(From pp. xi-xii of the draft of Invitation to Nonlinear Algebra by Michalek & Sturmfels)