What is the name of the element/structure that provides anchorage for thin
filaments and elastic filaments (these are pulled toward one another during
contraction)?

The Z- band is the element / structure that provides anchorage for thin filaments and elastic filaments.

Explanation:

The Z-band or Z-line is a narrow dark band that delineates the two lateral borders on a sarcomere and seperates the sarcomere, the smallest functional unit of a muscle.

These Z- bands are composed of many proteins macromolecules like α-actin, titin, nebulin etc. The thin actin filaments are attached to Z-band and to titin which are the protein macromolecules.

The titin protein also anchorage and maintains other bands in position on a myofibril.  Based on the sliding filament model of muscle contraction, the two Z-lines on a sarcomere comes close together during the sliding movement of thick and thin filaments.