The theory of Toeplitz quantization presented in our previous paper is extended and further developed to include diverse and interesting noncommutative realizations of the classical Euclidean plane. This is done using Hilbert spaces of entire functions, where polynomials in one complex variable form a dense subspace. The complex coordinate naturally acts as an unbounded multiplication operator generating, together with its adjoint, a highly noncommutative *-algebra of operators. The Toeplitz operators are then geometrically constructed as special elements from this algebra; they are associated to the symbols from another quadratic noncommutative algebra, which is interpretable as polynomials over a plane to be quantized. Such a conceptual framework promotes interesting nontrivial conditions on the initial scalar product. These are analyzed in detail. Various illustrative examples are computed.