Cell division is a natural process that involves no consciousness or action of the will. If I'm not mistaken, when it comes to single-celled organisms, the new cell will have all the same DNA of the original.
So it's a very bad analogy to what happens when countries break up. The two new countries could be very different. People will have different interests and attitudes and may end up in a country they don't want to live in.
If you have to use that analogy, it better fits the creation of colonies. The Greeks or the Phoenicians or the English could send out part of their people to a new city or province that aspired to be a copy of the original city or country.
Secession is a lot messier.
A Constitution copied from the USA. Legal Slavery. A President. A Legislature. Even a flag that looked like an American flag.
What would have been different?