You are asking about the Brandt 60mm and 81mm mortars. These were both designs of the French Brandt company.
The first design was the Brandt 81mm model 1927 mortar which was the standard battalion mortar of the French Army. Starting in 1925, the French engineer Edgar Williams Brandt redesigned and improved upon the World War 1 British Stokes 3-inch mortar using larger and more powerful mortar shells. The 81mm mortar can be easily assembled and disassembled in less than one minute so it only has three parts (base, tune, and adjustable firing stand) and can carried by three men anywhere on foot. The Brandt 81mm can fire either a 3.25 kilogram bomb out to 1900 meters or a 6.9 kilogram bomb out to 1000 meters with a maximum firing rate of 18 mortar rounds per minute.
The French government was bankrupt by World War 1 so the French government made the dumb mistake of licensing the Brandt mortar out to everyone who had money. With the exception of the British and the Commonwealth nations which had the Stokes Mortar; everybody else bought the Brandt design. So in WW2, you had the United States Army firing their improved version of the Brandt 81mm mortars at the Japanese, Italians, and Germans with the Axis forces firing back at the US forces with their licensed versions of the Brandt 81mm mortars. And yes, the ammunition is interchangable so the Germans could use US 81mm mortar ammo and the Americans can use captured Japanese, German, and Italian 81mm mortar ammo. The exception to this rule is the Soviet Union which enlarged their Brandt design to 82mm so the mortar can also fire captured 81mm mortar shells but no one else can fire captured 82mm mortar ammunition because the round was 1mm larger than the tube. The Britsh did use the Brandt design to improve their Stokes mortars to create the standard Ordnance 2-inch and 3-inch mortars that they and the Commonwealth nations used on the battlefield in World War 2.
The Brandt 60mm model 1935 became the standard company mortar for the French Army. The United States and Nationalist China also bought the license to produce the design. However, other nations by 1935 had used their licensed 81mm Brandt mortar licenses to create their own light mortar designs for their own militaries and for foreign sales so the 60mm caliber mortars were not that common. The Italians used a 45mm mortar design. The Germans, Russians, and Japanese used a 50mm design, and the British used a 2-inch Standard Ordnance design so there are a wide variety of light mortars used in WW2.
Yes, they did. To quite a large extent actually.
If you want to read about the Finnish use of mortars FINNISH ARMY 1918 - 1945: MINE THROWERS AND MORTARS is a good place to start