Aims. We study kink and fluting oscillations of a straight magnetic tube with an elliptic cross-section and density varying along the tube. Methods. The governing equations for kink and fluting modes in the thin tube approximation are derived. We found that there are two kink modes, polarised along the semimajor and semiminor axes of the elliptic cross-section. We have shown that the ratio of frequencies of the first overtone and fundamental mode is the same for both kink modes and independent of the ratio of the ellipse axes. Results. On the basis of this result we concluded that the estimates of the atmospheric scale height obtained using simultaneous observations of the fundamental mode and first overtone of the coronal loop kink oscillations are independent of the ellipticity of the loop cross-section.