Retain similarity distances when using an autoencoder for dimensionality reduction
No it is not possible to preserve relative distances when reducing dimensions for arbitrary data. This is not due to a property of auto-encoders compared to e.g. PCA or T-SNE. It is due to geometry.

You can see this relatively easily by considering a reduction of dimensions from 3 to 2, and examining a tetrahedron where all four corner points are 1 unit apart. There is no two-dimensional shape that can place four points mutually equidistant (unless perhaps you consider non-Euclidean spaces). It should be clear that relative distances near those corner vertices would also be affected, and thus this special shape demonstrates a general property of dimension reduction.