What is this tar like substance seaping out of the window caulk?

Do you live in an older home? They used to use tar paper many, many years ago to seal walls. Could be that it is melting and seeping out ... that's my guess. Another possibility is an old, large bee honeycomb inside the wall. Have you looked inside your attic directly this problem area?

Did you use this caulk anywhere else and is it doing the same thing in any other areas where you used it?

That is so weird! I read that someone else had a problem similar to this and they thought it was honey and tasted it (barf), but that it turned out to be a racoon nest and all the feces had finally started draining out of the walls. I don't think that's the case with your house, but let us know when you find out what it is!

This black goo looks like the peel-and-stick flashing they put around windows. I assume this is the tar or mastic that was melting from the excess temperatures they are exposed to.

If this is some kind of sealant that was used instead of a proper flashing job then this may well be a failure in the making.

I do know those rubberized-asphalt flashings should not be in direct contact with flexible vinyl flashings or caulking.

From the look of the sun exposure in the photos, it may be just those vinyl flanges reacting exposed to the solar and heat.

It is difficult to do any detective work without knowing all the facts. However, you do! So have a think and see if any of the answers above fit in. How old is your home? Has the temperature been extremely hot or cold this year - or wet or dry? Do you know how the house was constructed? Is there a crawlspace? Is this a first story window? What is the outside covering - brick, stucco, wood? Does this stuff have an odor? How viscous is it? Narrow down the possibilities and you'll find a fix.

Stop guessing; call in a professional!!

I agree that the black goo looks like the peel-and-stick flashing they put around windows.

I recommend trying WD-40.