No it isn’t. The metaphor is that if you throw a frog into already boiling water it would attempt to jump out. If you start with tepid water and increase the temperature slowly enough they don’t. Sadly this was proven through experiment in the 1800s.
It’s an argument that if you make changes slowly enough people won’t notice.
In the experiment you mention, before they put the frog in the cool water, they removed its brain. Then they boiled the water. The frog did not jump out of the water because it had no brain. The experiment proved the opposite of what you are asserting.
If every wealthy country had a frog to represent their culture of taking care of workers (strong unions, workers rights, vacation days, not having healthcare tied to their employment, maternity and paternity leave, equitable pay etc), there is one particular frog which most would describe as having had its brain removed.
From the wikipedia article linked to just below this reply, it says that the first such experiment is as you described. But then goes on to say:
Other 19th-century experiments were purported to show that frogs did not attempt to escape gradually heated water. An 1872 experiment by Heinzmann was said to show that a normal frog would not attempt to escape if the water was heated slowly enough, which was corroborated in 1875 by German scientist Carl Fratscher.
I don't see the point of the experiment with the brain removed, but given that they did the experiment with intact frogs as well confirms their original hypothesis.
However, later on in the article, it's been disputed in recent years: as the water is heated by about 2 °F (about 1 °C), per minute, the frog becomes increasingly active as it tries to escape, and eventually jumps out if it can. Earlier it also says that frogs put into already water just die (not mentioned, but presumably from shock) and so don't have a chance to start attempting to jump out. I imagine humans dumped into boiling water would have a similar response.
