Not sure what you find odd about it - it’s just not part of your praxis. It is an Eastern liturgical object, a small mat or rug placed where the bishop stands during the services. It shows an eagle flying over a city, symbolising the bishop overseeing his diocese - which, basically, is the meaning of the word episkopos.
Commonly it is called an ‘eagle’. There is often a small army of servers or subdeacons rushing to place one wherever the bishop is headed at various times during the service (not so much a problem in cathedrals which will usually have a whole pile of eagles).
At the beginning of an episcopal consecration a much larger one is used and the one to be consecrated is led first onto the tail, where he makes a profession of faith, then onto the body of the eagle, where he makes a further detailed profession of faith, and finally he is led onto the head of the eagle where he undertakes to observe and maintain the canons of the Church.
In former days kings and emperors stood on a rug or mat of silk or sable furs - particularly at their anointing and coronation,
Even today, in the Russian tradition at least, the bride and groom similarly stand on a silk mat or rug. There are various traditions (superstitions, really) concerning who should step on first - which might decide who will be head of the household - and the consequences of one or other of the couple turning their corner over as they step on to the rug (accidentally or deliberately!).
I don’t know which or any of the Eastern Catholic Churches use the eagle rug.
