Are rats really the first to leave a sinking ship?

Animals have pretty good instincts, they got to high ground well before the tsunamis hit a few years ago, so probably yes.

Rats live down at the bottom of a ship, so when a ship is taking on water, the rats seek a higher, dry place. So when a ship is sinking, one sees the rats coming up. And they are the first to know that water is collecting in the holds.

Along the lines of what @zenvelo says, pirates would insult you by calling you a bilge rat.
The bilge being the foulest/darkest part of a ship located beneath the floorboards on the lowest deck.

Sounds like a metaphor.

@gondwanalon It is a metaphor, but many a metaphor is based in reality.

Fleas and tickets rapidly decant from moribund animals (as the body temperature drops). A “rats leaving the sinking ship” sort of situation.