The wire is electro-galvanized steel. Please note that not all methods of galvanization are "equal" when it comes to posing a risk of zinc ingestion.
The two main methods of galvanization are hot dip galvanization and electro-galvanization:
• Hot dip galvanization is where the metal is dipped in essentially a bath of molten zinc. Metal treated in this fashion has a dull, rough and flaky surface of zinc. If a bird spends an inordinate amount of time chewing on toy hardware with a flaky galvanized coating then it is possible they could ingest enough zinc to build up toxic levels in their system over time.
• Electro-galvanization is process where a thinner, tighter-bonding coat of zinc is applied to a metal via electroplating. The surfaces of the materials treated by this method are smooth and sometimes shiny in comparison to those treated by the hot dip method. In other words, there are no loose or flaky zinc components available for birds to swallow. Metals treated by this procedure do not pose a risk of zinc ingestion as long as the metal object cannot be swallowed in its entirety.
Many avian health specialists have stated that in order for zinc related health problems to result, a parrot must actually swallow the metal components and that "beaking" the parts is insufficient for toxicity to develop.
Swallowing the wire used within the bungee would not be physically possible and therefore poses no risk.