Let \(\mathbb{T}^d\) be the \(d\)-dimensional torus, let \(X\) be a normed linear space of real functions defined on \(\mathbb{T}^d\), and let \(C(\mathbb{T}^d)\) be the space of all real, continuous and periodic functions on \(\mathbb{T}^d\). In order to compare different approximation algorithms that are applicable to the functions belonging to a given set \(K\subset C(\mathbb{T}^d)\cap X\) and are based on standard information, i.e. the values of the involved functions at \(n\) points chosen from \(\mathbb{T}^d\), the \(n\)th minimum intrinsic error \(e_m\) and the \(n\)th minimum linear intrinsic error \(e^L_n\) are relevant. In the present paper the authors estimate these errors in the case when \(X:= L^{{\mathbf s}}_{{\mathbf q}}(\mathbb{T}^d)\) and \(K\) is the unit ball of the anisotropic Besov space \(B^{{\mathbf r}}_{{\mathbf p}\theta}(\mathbb{T}^d)\), where \({\mathbf s}: =(s_1,\dots, s_d)\), \({\mathbf q}:= (q_1,\dots, q_d)\), \({\mathbf r}= (r_1,\dots, r_d)\) and \({\mathbf p}:= (p_1,\dots, p_d)\) are vectors satisfying \(s_i\in \mathbb{Z}_+\), \(1\leq q_i\), \(1< r_i\), \(1\leq p_i\) for \(i:= 1,\dots, d\), and \(\theta\) is a number satisfying \(1\leq\theta\leq\infty\). By using the Dirichlet approximation algorithm they reveal cases (depending on \({\mathbf p}\) and \({\mathbf q}\)) when the asymptotic equalities \[ e_n\asymp e^L_n\asymp n^{-g(r)(1-\beta)}\quad \text{and} \quad e_n\asymp e^L_n\asymp n^{-g(r)(1-\beta)+ g(r)\gamma} \] hold, where \[ \begin{aligned} g(r)&:= 1/\Biggl(\sum^d_{i=1} 1/r_i\Biggr),\\ \beta&:= \max(s_1/r_1,\dots, s_d/r_d),\;\gamma:= \sum^d_{i=1} (1/p_i- 1/q_i)1/r_i. \end{aligned} \]