This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic systems involving nonlinearities with Trudinger-Moser exponential growth. We first study the existence of solutions for the following system: $$\begin{aligned} \left\{ \begin{array}{ll} -\big (a_1+b_1\Vert u\Vert ^{2(\theta _1-1)}\big )\Delta u= \lambda H_u(x,u,v) &{} \quad \text{ in }\ \ \ \Omega ,\\ -\big (a_2+b_2\Vert v\Vert ^{2(\theta _2-1)}\big )\Delta v= \lambda H_v(x,u,v) &{} \quad \text{ in }\ \ \ \Omega ,\\ u=0, v=0 &{} \quad \text{ on }\ \ \ \partial \Omega , \end{array}\right. \end{aligned}$$where \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with smooth boundary, \(\Vert w\Vert =\big (\int _{\Omega }|\nabla w|^2dx\big )^{1/2}\), \(H_u(x,u,v)\) and \(H_v(x,u,v)\) behave like \(e^{\beta |(u,v)|^2}\) when \(|(u,v)|\rightarrow \infty \) for some \(\beta >0\), \(a_1, a_2>0\), \(b_1, b_2> 0\), \(\theta _1, \theta _2> 1\) and \(\lambda \) is a positive parameter. In the later part of the paper, we also discuss a new multiplicity result for the above system with a positive parameter induced by the nonlocal dependence. The Kirchhoff term and the lack of compactness of the associated energy functional due to the Trudinger-Moser embedding have to be overcome via some new techniques.