Circular black rubber part inside MacBook Pro - what is it?

Not sure. I'd check the illustrated repair instructions at iFixit and look for something familiar in their image, which when clicked on will enlarge at a usable resolution.

https://www.ifixit.com/Device/MacBook_Pro_13%22_Unibody_Mid_2010

Try several of the guides listed to get different views of the interior parts.

it appears to be a drive spacer which - if it is - is used to secure the HDD to the chassis so the drive doesn't' strike or bump any hard surface when the laptop is moved. It's not any part of the logic board AFAIKT. Also if it is, it's possible the factory used one type that doesn't look like "stock apple" then ran out and used whatever they had on hand so production wouldn't come to a halt, it's also possible the drive was changed out and someone forgot to take out the spacer.

I don't believe the 'use whatever part is to hand' theory of Apple production, it was more likely to be a part supplied by a non-Apple source, like a replacement HDD or SSD.

I have 7 MBP's, none of them possess that part.

And Apple place their drive shock absorbers in the mount for the drive, nothing just lays around randomly.

It may not have come directly from Apple and I never stated it did, however it's not impossible. My understanding is at one point Apple had parts coming from two different factories in Taiwan. Likely this is a 3rd party f up but we had a Buick with a coke bottle welded in the front quarter panel at the factory so it's not impossible something or someone happened at the factory to cause a problem which was not evident in 7 other buicks owned by anyone else.

Looks like a shock absorber used when installing the internal drive. Would be four of them.

Was this a new model, or used.

It doesn't look like an Apple part, maybe supplied by a drive manufacturer?

No more guesses, just a suggestion, don't throw it away yet. 🙂

Fortunately Macs are built better than Buicks.