When playing a 1080p 3D blu-ray video, is the image quality actually lower?

Theoretically, yes, but in practice it's more complicated than that.

First of all, we are going to only talk about passive 3D. That's where the glasses you wear to watch it have no electronics or connection to the TV.

In order to make passive 3D work, the information for one eye is displayed on the odd lines of the TV and the information for the other eye is displayed on the even lines. That means on a 1080p TV, each eye is getting 540 lines (instead of 1080). So that would be half the vertical resolution, although our eyes tend to interpolate so the apparent resolution is more likely somewhere between 540 and 1080 lines. By that I mean, if you watched a 2D film with 540 lines, it would probably look worse than passive 3D 1080p.

There's another wrinkle, though, which is that better 3D TVs worked out a way around the limitation of passive 3D. What is done there is that the refresh rate of the TV is made at least double that of the incoming frames from the media. Then the full 1080 lines for each eye are displayed in two batches of 540 in the same amount of time as a single frame. Basically, each eye is interlaced. That makes each eye's approximate effective resolution to be 1080i, with the normal interlacing artifacts.

At least as of 2014, some 4K TVs supported passive 3D, and yes, those TVs would present 1080 lines to each eye, since they have double that number of total lines of resolution (2160).