Kind of on Topic

[StainlessSteelCynic]

Armies that have been influenced by British tradition typically have their designations meaning the 2nd and the 4th Cavalry Regiment rather than the 2nd sub-unit of the 4th Cav Regiment. Note also that full unit names can include brackets i.e. ()
But to go further with what both Legbreaker and Rainbow Six have mentioned, the designations are also used to "clear up" some of the messiness that could result when units are amalgamated or designated as successors to previous units.

Successors are exactly that, sometimes a unit will be disbanded but its designation will be incorporated into the designation of a newly formed unit so that there is a historical link to the disbanded unit and it's battle honours are not lost.

In regards to the "messiness", I'll use 1/15th Royal New South Wales Lancers as an example as this unit has a complicated lineage.
The core of 1/15th RNSWL was originally the New South Wales Lancers, a colonial cavalry [recce and skirmishing] unit formed in 1885. During WW1 it was retasked to mounted infantry and renamed the 1st Light Horse Regiment.

After WW1 it was amalgamated with the 21st Light Horse Regiment to become the 1st/21st Light Horse Regiment (New South Wales Lancers).
In 1935 the unit received the prestige title "Royal" and the link to the 21st Light Horse was uncoupled in 1936 and later the unit converted to a motorised machinegun unit designated the 1st Light Horse (Machine Gun) Regiment (Royal New South Wales Lancers).

During WW2 a new unit, the Second Australian Imperial Force AKA 2nd AIF, was raised from volunteers to serve overseas - at that time, militia units such as 1st Light Horse could not serve outside Australian territory/protectorates due to the Defence Act of 1903. However a very large portion of 2nd AIF were volunteers from 1st Light Horse. 2nd AIF was assigned to a Machine Gun Battalion and was later renamed the 1st Machine Gun Regiment.

In March of 1942 the 1st Machine Gun Regiment converted to a vehicle role and became 1st Motor Regiment however in May it converted to Matilda tanks and became the 1st Tank Battalion of the 3rd Army Tank Brigade. After being relisted as part of the regular forces and then deployed to the western Pacific to fight the Japanese, the units actions helped prove the suitability of tanks for jungle warfare.

In 1944 the unit was renamed as the 1st Armoured Regiment but after the war ended, the unit was demobilized and thus was no longer a unit within the Australian Army.
In 1948, reserve forces (known as Citizens Military Forces AKA CMF) were re-established and the demobbed 1st Armoured Regt was reborn but as a CMF unit with the name 1st Armoured Regiment (Royal New South Wales Lancers) in recognition of the history and battle honours of the earlier unit.

But in 1949, the unit was redesignated as 1st Royal New South Wales Lancers so that the name 1st Armoured Regiment could be allocated to a new armoured formation in the newly reorganized and renamed Australian Regular Army.

In 1956, things got a little less messy because meanwhile...

The 15th Light Horse Regiment (AKA 15th LH), was raised in Palestine in 1918 from members of the Australian Battalion of the British Imperial Camel Corps Brigade. After the war the unit was disbanded but re-raised as a CMF unit in the Northern Rivers region of New South Wales and linked to the 5th Australian Light Horse Regiment (New South Waled Mounted Rifles), a unit raised in 1903.

In December 1941 15th LH was converted to motor vehicles and renamed the 15th Motor Regiment (Northern Rivers Lancers) in recognition of their previous history. In 1942 the unit gazetted as an AIF unit (so that it could be deployed to warzones outside Australian territory) and was again renamed to 15th Australian Motor Regiment. However the unit was deemed unneeded and disbanded.

In 1948 with the re-raising of the CMF, the disbanded 15th was reformed as a squadron focusing on riverine operations (due to their location in a region with many waterways), and named as A Squadron, 15th Amphibious Assault Regiment (Northern River Lancers). In 1949 the name was simplified to 15th Northern River Lancers.

Which brings us back to 1956 where as part of the reorganization and rationalization of army units...
1st Royal New South Wales Lancers and 15th Northern River Lancers were amalgamated to form 1/15th Royal New South Wales Lancers.

So yeah, Australian designations are quite different to US designations and they follow the British pattern of keeping historical and traditional links to earlier units or to units that are similar (although sometimes units are combined for simplified peacetime admin only to be split apart again in wartime).