It shouldn't. Not that I have ever seen. Is yours leaking? I could be wrong. I never tore in to one to find out, but I always thought it was no oil.
The same mount location on my TSX and TL had the oil filled type. Those look different though.

 

50800-S3V-A82 mount at radiator side of engine can be hydraulic.

50800-S3V-A82 - Rubber Front Engine Mounting - 2003-2008 Honda Pilot | Honda Factory Parts

 

Look at the post below. Can you trace the oil back to maybe a spill when filling the engine oil? Or power steering fluid since the hose and pump are in that area. The o-ring for the power steering line harden and crack over time. Amazon.com : honda pilot power steering o ring These are o-rings for the pressure hose.

 

The leak (dark brown) and defiantly NOT Power Steering fluid, oil or hydraulic dripping from around rubber of p/n 50820-S9V-A01 engine mount (passenger side) I am very anal in keeping the engine clean (you could eat off it) I was hoping someone knows if the mount is oil filled. One repair shop says no and another says yes. Other than the mount manufacturer who would know ? Is there a contact to confirm this?

 

I had a 2003 Pilot. Had leakage below the passenger side motor mount. Had it replaced, no more leakage. Don't know about 2005.

 

I have seen them fail, and never seen oil in them. Doesn't mean they don't have oil in them, but the chances are very slim. The one that does is shown in the picture above. Notice how it has a metal housing for it. This is because the rubber flexes, and micro cracks, and if they filled it with oil, I don't even see how it would benefit the design. There is about a 1 percent chance there is fluid in that mount. That just just because I haven't actually dissected it.
Could the oil you see be coming from when you cleaned it, and it pooled somewhere, and discolored from the rubber, and finally got squeezed out after flexing? LOL. You never know. Touch it and rub it in your fingers. Is it water or oil?

 

Haven't had the side mount in our 05 Pilot fail while it's been under my watch, but the one in our old 05 Odyssey and 16 Odyssey both failed (tore) and they were definitely fluid filled - left brownish stuff down the frame rail, etc. - they look pretty similar to the side mount in the Pilot, but that certainly doesn't mean the pilot's side mount definitely is. What came out of those was more watery than oily, though. (and after removing them, they were definitely "hollow" where the fluid used to be.

 

Found it.

 

Further proof, enter ( "engine mount" & "fluid") into the search community box / advanced search and you will see posts going back to 2003 model about failing mounts with leaking fluid.