Note that, there was no formal treaty or formal cease fire and negotiated end to the war in 1866 either (as far as I know). Andrew Johnson just said "the threat is gone, the war to be over."
Of interest to me is that if you look at his "the war is over proclamation," Johnson could have issued it months earlier, given the conditions on the ground. There's probably a story behind why he happened to issue the proclamation in August 1866, but I don't know it.
Johnson returned the country to normal with various proclamations over several months...
May 10, 1865 declaration that armed resistance was virtually over and ordering arrest of insurgent cruisers.
May 22, 1865 declared the blockade on ports lifted except Texas.
May 29, 1865 granted amnesty to participants in rebellion with certain exceptions.
May 29 - July 13, 1865 individual proclamations to reorganize constitutional government in NC, MS, GA, TX, AL, SC, FL.
June 13, 1865 removed trade restrictions on former Confederate states east of Mississippi River.
August 29, 1865 removed restrictions on contraband of war.
October 12, 1865 declared end to martial law in Kentucky.
December 1, 1865 Restored habeas corpus in all states except Kentucky and the eleven states of the Confederacy.
April 2, 1866 declared insurrection to be at end except for Texas.
August 20, 1866 declared insurrection over even in Texas
September 7, 1867 offering full pardon to all persons participating in Late Rebellion with some exceptions.
October 7, 1867 correction to the August 1866 proclamation declaring insurrection over.
Johnson waited on the seceding states to form state governments under the presidential reconstruction plan before declaring the insurrection over. Texas was last on the list with its new constitution ratified and the new state legislature met on August 6, 1866. However, the state governments were removed in 1867 by Congress and reconstituted as military districts with eventually a state government reconstituted by congressional rules (except Tennessee).
So the 8/20/1866 date is weak since Johnson's motivation for declaration was the establishment of civilian governments in the seceded states which were soon swept away. The shooting war ended in 1865. As noted above he reissued the "Insurrection is Over Proclamation" in 1867 because of incorrect dates in the original.