The colonization of Liberia started with the creation of the American colonization society (ACS) who was an organisation devoted to resettling free people of colour to Africa.

This organisation was supported both by abolitionist (like the Quakers) who believed that African Americans would have faced less oppression in a free state of their own and slaveowners who feared that free PoC would incitate slave rebellions.

This organisation financed the emigration of PoC to Africa (framed as a return to their homeland) starting in 1822, and migration got more intense after the civil war.

In 1847 Liberia declared independence and was recognised by several European nations, which later pressured it to give up most of its claims in the area.

And that's why the flags are so similar: Liberia was colony of the United states, founded to resettle PoC in a mew land, and later became independent and adopted a similar constitution, government and religion to that of the united states, from where the ruling class came from.