I took a closer look at the original model and noticed that you have a Frictional Contact with some friction between the inner and outer tubes and you have used Adjust to Touch. Was that intentional? I hope you understand that means the contact elements that cover the surface of the small pipe are offset from the pipe surface until they touch the inner diameter of the large tube, therefore the small pipe is already supported by the large tube before any load is applied.
Read the ANSYS Help on Eigenvalue Buckling. Eigenvalue Buckling Analysis (ansys.com)
Since you have a nonlinear item in your Static Structural model, the Eigenvalue Buckling is using the Linear Perturbation Analysis procedure.
I also notice that there are two loads on the static structural model: the force and gravity.
The load multiplier is for both loads, not just the force. You would have to replace Standard Earth Gravity with Acceleration so you can apply the multiplier to the acceleration of 9.81 m/s as well as the force.
Eigenvalue Buckling is a linear analysis and so is Modal analysis. In Modal analysis, an automatic conversion of nonlinear contact to linear contact is performed, so a Frictional contact that is closed (as yours is due to Adjust to Touch) would be converted into Bonded contact. I don't know how Eigenvalue Buckling treats frictional contact. Perhaps someone who knows can answer.