We show that the system can be tuned into three different regimes of effective qubit-qubit interactions, namely a transverse (XX+YY), a longitudinal (ZZ) and a non-trivial ZX interaction. In addition, we show that an enhanced coupling can be achieved by employing an ellipsoidal magnet, carrying anisotropic magnetic fluctuations. We propose a scheme for realizing two-qubit gates, and simulate their performance under realistic experimental conditions. We find that iSWAP and CZ gates can be performed in this setup with an average fidelity ≳99% , while an iCNOT gate can be applied with an average fidelity ≳88%. Our proposed hybrid circuit architecture offers an alternative platform for realizing two-qubit gates between superconducting qubits and could be employed for constructing qubit networks using magnons as mediators.