Early Roman standards, which preceded their use of the eagle, were simply bundles of straw attached to the end of the pole. Could it possibly be that? The Romans then moved on to using variations of the eagle, the wolf, the minotaur, the horse, and the wild boar, before finally settling on the eagle (known as the aquila), using that bird of prey as the recognized standard by the Roman Legion. According to Pliny the Elder, “Caius Marius, in his second consulship, assigned the eagle exclusively to the Roman legions." Marius was the dude who apparently abolished the use of the other standard displays in full favour of the eagle. Pliny also referred to the eagle as the “most honorable and strongest of all birds”, and it was clear the Romans believed that the eagle was equal to the mythical dragon in terms of its power, so they clearly thought a lot of the bird. Heck, the German Nazi party and the Americans adopted it as one of their primary symbols as well, so it has long been seen as a symbol of strength and power.

As for its perch you are asking about, I know the eagle was typically represented by the Romans (just as you said) as holding a thunderbolt (or what they called a fulmina) in its claws - making it the armament and armour-bearer of Jupiter - many times with the letters SPQR (Senatus Populus Que Romanus) below in order to praise the Senate and people of Rome.

The object we see the eagle perched on in your photos there is clearly not looking like a typical thunderbolt/fulmina shape we would normally, in modern context, relate to. I have seen examples/reproductions of the Roman eagle sitting on the same perch as you show, but with an addition to the perch, as seen here for example. In this case, you can clearly see the "thunderbolt"/lightning bolt shapes added to the ends of the perch. Here is another version, showing the same "thunderbolt" style perch, though in this case the thunderbolt is above the actual shape you are asking about. Admittedly, whether this reproduction version is accurate at all for the ancient Roman time period is speculative at best though, because the fact is that there are very few contemporary examples of complete Roman eagle statues/standards that have ever been found. Plenty of recreations exist of course, but not much from that time period to go on.

There are a bucketload of Roman coins that have been recovered which feature the eagle stamped on one side of the coin though, and those coins vary in imagery, from showing the eagle perched on what appears as a simple staff/stick, to the eagle perched on what is believed to be a palm branch, to the bird's talons grasping what is described, even if in some cases it doesn't really look like it to our modern eyes, as being a "thunderbolt".

So, after my long-winded and unfortunately not too informative response to your specific question, I really don't think there is any consensus as to what the heck that eagle is actually perched on in the pics you linked to, and no firm understanding of what that shape is supposed to represent. Again, there are examples of that object showing what are appreciated by many as being lightning/thunderbolts emanating from it or around it, but as for what that perch really is supposed to represent, I confess I do not know. I am hoping that others here might be able to offer some insight and evidence to answer that, as it is a good question! Again, sorry for the rambling, but I would love to hear from others far more schooled in this subject matter than I if there is any firm evidence that can help answer your query as well.