The method used in one low-end "dumb" domestic use video switching system I have seen, consists of shutting down the video signal very briefly to a blanking signal, then switching on the second source. This brief blank period of a few frames allows the receiving device to resync to the new source, without the frame glitches mentioned.
No, something like this is not typically a single-IC solution. It takes a bit more than that.
For more sophisticated systems, as mentioned in other answers and in comments, the switching device needs to have a frame buffer and mix in the two source signals. This also allows the cross-fade transition between sources (and, optionally the picture-in-picture features) that some AV modular receivers offer.
Note though that for a simple frame-buffer based transition or switching system, the two DVD sources must have the exact same resolution and output format: Just having identical DVD players is insufficient, the source DVDs being played must be of the same audio and video specifications.