Tying everything said in various answers and comments together: Shimano calls the item in question an e-ring, but Kibbee says e-clip may be the more general term. As stated by Kibbee and Criggie, e-rings usually hold something like a post in place. And it appears that the e-ring is linked to a post that holds the lever blades in place (these are the things you pull to brake, or push to shift).
While it doesn’t appear to be documented in the replacement parts diagram for the R7025 shifters, Shimano does still sell the clips and posts as replacement items. I guess this would be useful if you broke the lever blades. The blades are also available as replacement parts, but they seem pretty expensive, and if you're in this situation it could be worth scouring eBay for single STI shifters or ones that have broken from other causes. The picture below is the replacement post and axle for the 8000 series, sold by SJS Cycles in the UK. It claims compatibility with 7000 and 9100. There are different listings for other models of STI shifters, e.g. here is the one for the 6800 and similar era units. Shimano also sell a specialized removal tool to fit the e-ring.
A manual for the 7900 STI levers (Dura Ace, second 10 speed generation) documents how to remove the post. The pictures in question are under the maintenance section. You'd use the special tool I mentioned earlier (also pictured in the image shown by Dan K, also shown in the manual I linked) to push the e-ring off the post. However, end users should not have to do this regularly. The shifting internals in STI levers certainly don't need regular maintenance. They will eventually wear out, but that should take a while. The internals are very complex and have very small parts, and are not generally possible to service by hand.
It does not appear that any of the parts involved in bleeding the brakes would involve getting near the shift lever blades, or that the bleed process involves moving a different post or an e-ring. A scan of the bleed process documented in the dealer manual seems to confirm this. Fortunately, the OP confirmed that both their STI units had the e-rings present. The origin of the pictured e-ring must remain an eternal mystery.