What two countries have the same name?

American Samoa and Samoa.

Republic of the Congo and the Democratic Republic of the Congo.

British Virgin Islands and the US Virgin Islands.

The Republic of China and the People's Republic of China.

Dominica and the Dominican Republic.

Ireland and Northern Ireland.

Guinea, Equatorial Guinea, and Guinea (Bissau) - and you can also include Papua New Guinea, New Guinea, Cameroon and Togoland which were formerly German Guinea, and Ghana which was formerly Dutch and Danish Guinea.

Guyana, French Guiana, and Suriname, formerly known as Dutch Guiana.

Honduras and Belize, formerly known as British Honduras.

North Korea and South Korea.

A more recent example: Sudan and South Sudan

Some that have similar names or heavily related etymologies are:

Grenada, and St Vincent and the Grenadines.

Austria and Australia.

The collective Guineas and Guianas/Guyanas.

Niger and Nigeria.

Somalia and Somaliland.

India and Indonesia.

Turkey and Turkmenistan.

Albania and Scotland, also known as Alba.

Sealand and New Zealand, as well as possibly the Zeeland region of the Netherlands.