During the two world wars, would commanders on the ground have known the exact name of the regiment, division or army group?

For example I was reading about Operation Totalize, The 1st Canadian (Armoured) Division and the 1st Polish Armour Division were fighting 12th SS Panzer, 101st SS Heavy Panzer and 89th Infantry Division.

Would commanders have known it was the 12th SS Panzer, 101st SS Heavy Panzer and 89th Infantry

Yes. During World War II, the U.S. Army Military Intelligence Service and Military Intelligence Division (MIS/MID) made a prodigious effort to gather intelligence on the battlefield about enemy troop disposition, strength, and morale, and how best to exploit them to their advantage. Various teams of intelligence personnel were assigned or attached to field armies, corps, or divisions. IPW (interrogation, prisoner of war) or PWI teams gathered information from captured prisoners, such as their names, service numbers, the names, types, immediate disposition, and strength of their units and fortifications, as well as the general state of enemy morale. OB (order of battle) teams attempted to construct orders of battle (i.e., "X" number of volksgrenadier divisions, "Y" number of panzer divisions, and "Z" number of panzergrenadier divisions subordinated to the "123rd Corps," subordinated to the "456th Army") from captured enemy prisoners and documents, while PI (photo interpretation) teams analyzed information gathered through aerial and ground photoreconnaissance. MII (military intelligence interpreter) teams interrogated civilians in order to find out information on enemy units that could have been inadvertently exposed, such as unit names, locations, strengths, and defenses constructed, and the general state of enemy morale.

The PI and MII teams nominally had one captain, one first lieutenant, one master sergeant, and two technicians third grade, while the IPW team had one captain, one first lieutenant, one master sergeant, one technician third grade, and one technician fifth grade, and the OB team had only a captain, first lieutenant, and technician third grade. In the European Theater, normally, each division had two IPW teams (shuffled around as needed depending upon how big the prisoner bag was), one OB, one MII, and one PI team attached. Each corps headquarters had three IPW teams, two MII teams, four PI teams, and one OB team. Each field army headquarters had four IPW teams, three MII teams, four PI teams, and one OB team. 12th Army Group headquarters had two MII teams, five PI teams, and one OB team. European Theater Communications Zone headquarters had 12 IPW teams and 33 MII teams at their disposal.

The Counterintelligence Corps (CIC) had operations both in the United States and abroad. Cooperating with the FBI and Provost Marshal General (the highest-ranking military police officer and commander of the Corps of Military Police), it conducted background checks on personnel who had access to classified information, and investigated sabotage, espionage, subversion, or allegations of disloyalty. Overseas, CIC detachments informed friendly troops about the danger of enemy sabotage, espionage, and subversion, and actively hunted for enemy spies and saboteurs behind friendly lines. Nominally, there was one CIC detachment attached to each division.

On a more local level, intelligence was carried out by the G-2 (general staff, intelligence) sections of divisions, located in the general staff section of the division headquarters in the infantry division, and the "forward echelon" of the division headquarters of the armored division.

In the infantry division, each infantry regiment had its own intelligence officer (S-2), as well as an intelligence and reconnaissance (I&R) platoon in its headquarters company, which essentially served as the regimental commander and S-2's personal intelligence service. The intelligence and reconnaissance platoon consisted of a platoon headquarters of one officer and six men with a jeep, and two squads, each of nine men and three jeeps. The intelligence and reconnaissance platoon could be used to conduct mounted or dismounted patrols, probe enemy lines and capture prisoners, establish observation or listening posts, warn higher headquarters or adjacent units of enemy movements, maintain contact with adjacent units, or in emergency cases, be plugged into the line and be used as additional infantry. Each of the regiment's three battalions also had its own S-2 officer.

In the armored division, each of the three tank battalions had a reconnaissance platoon in its battalion headquarters company. It had a platoon headquarters of one officer and five men with a halftrack and a jeep, and two squads, each of eight men and two jeeps. Each of the three armored infantry battalions had a reconnaissance platoon in its battalion headquarters company; it was organized identically to the tank battalion's reconnaissance platoon. The tank and armored infantry battalions also had S-2 officers.