“[…] A great star burning like a lamp fell from heaven, and it fell on a third of the rivers and on the springs of waters. The name of the star is Wormwood.”
~Revelation 8:10,11

“And I saw a star that had fallen from heaven to the earth, and the key to the shaft of the abyss was given to him.”
~Revelation 9:1

Is the star representing the same thing or spirit in both passages?
There are so many “star” references in Revelation that it comes in handy to have a list of how the word is used in other Bible passages.

• Messiah (Num. 24:17)
• Jesus in heaven (Rev. 22:16)
• Kings descended from David (Is. 14:13)
• Angels (Job 38:7)
• Demons (Rev. 12:4)
• Holy ones (Dan. 7:25; 8:10,24)
• Acts of divine intervention (Jg. 5:20)
• Gloomy outlook (Is. 13:10; Ez. 32:7,8)

Revelation continues the sequence of events by calling Jesus “Apollyon,” which means Destroyer. (Rev. 9:11)
Wormwood represents bitterness, and a message of destruction would be bitter to those about to be destroyed.
So at the risk of being completely wrong, it seems logical to me to infer that the stars in both passages refer to Jesus.
We can also take into account a parallel passage which explains that the rivers and springs became the blood of the holy ones and prophets being poured out in God’s anger. (Rev. 15:7; 16:1,4)
It is common to equate bitterness with anger.
It does not seem that God would use a demon to execute an act of justice, but rather, he would use Jesus, as he usually does.
Additionally, the passage goes on to say a third of the sun, moon and stars were darkened, that neither the day nor night might have any light in them. (Rev. 8:12)
This is obviously symbolic because a literal application would wipe out life as we know it.
But this symbolism is in stark contrast to the star that burns like a lamp in its descent.
The passage seems to compare Christ’s enlightened message denouncing those who have persecuted God’s people with the spiritual darkness that they now find themselves in.