The specific type of bird represented in the picture is different, in that it does have live birth, so the joke works.

But it would still work regardless, because a joke doesn't have to be scientifically accurate - it can play off the human tendency to anthropomorphize, so a "big bird with little birds inside it" can still be a mother instead of the logical explanation that it's a predator, and this unexpected shift in view point (ruthless predator who just ate a bunch of babies, to the loving aspect of motherhood) is what makes it funny.

We're overanalyzing this...

Fresh eggs are still alive by some metrics, even if there may not be e.g. a heartbeat before they are layed.

Well I'm not sure what made that nest, but a chicken has about 5 developing eggs inside at one time. Each of which contains a tiny bird, according to the present SCOTUS.

To further undercut the joke, it's more likely the nest is old, the babies fledged and the cat is just relaxing.