The authors investigate some topological properties of self-similarity sets for families of some bi-dimensional maps. More specifically they consider the two-parameter families \[ T_i(x,y)=\left(\frac{x+i}{\beta_1},\frac{y+i}{\beta_2}\right), \; i\in\{\pm 1\}, \; \beta_1,\beta_2>1. \] The considered set is the unique compact set \(A=A_{\beta_1,\beta_2}\) such that \(A=T_1(A)\cup T_{-1}(A)\), i.e., \(A\) is the attractor of the iterated system \(\{T_1,T_{-1}\}\). The main purpose of this article is to study the interior of the set \(A=A_{\beta_1,\beta_2}\). It is proved that for a set of values of the parameters \(\beta_1,\beta_2\) the point \((0,0)\) belongs to the interior of \(A\). As a corollary of the main theorem it is proved that for \(1<\beta_1,\beta_2<1.02\) the origin belongs to the interior of \(A\). This improves an earlier result, [K. Dajani et al., Indag. Math., New Ser. 25, No. 4, 774–784 (2014; Zbl 1338.37018)], in which the above mentioned result was established for the range \(1<\beta_1,\beta_2<1.05\).