I claim the ornamental design for a drinking container handle, as shown and described.
FIG. 1 is a left side perspective view of Applicant's new drinking container handle design;
FIG. 2 is a front view of the drinking container handle design;
FIG. 3 is a left side view of the drinking container handle design;
FIG. 4 is a right side view of the drinking container handle design;
FIG. 5 is a top view of the drinking container handle design; and,
FIG. 6 is a bottom view of the drinking container handle design.
The broken lines showing a drinking container are for the purpose of depicting exterior and interior features of a drinking container and form no part of the claimed design.