This is unknown.
Much of the Osgiliath episode is a movie invention, so there's no source in the books to say whether Boromir would have used the same, a different, or even any shield at all when not with the Fellowship.
In the books Boromir's shield is actually not described at all (it's not even said to be round, which is commonly assumed, I believe, based on the Brothers Hildebrandt illustration of the Fellowship); the only thing that is said of it is in the description of Boromir's equipment when leaving Rivendell:
Boromir had a long sword, in fashion like Andúril but of less lineage and he bore also a shield and his war-horn.
Crucially, in the scene where Boromir's funeral boat is described, his shield is not included among the various items arrayed about him (despite this description being detailed enough to include the fact that Boromir also wore a helmet when with the Fellowship):
Now they laid Boromir in the middle of the boat that was to bear him away. The grey hood and elven-cloak they folded and placed beneath his head. They combed his long dark hair and arrayed it upon his shoulders. The golden belt of Lorien gleamed about his waist. His helm they set beside him, and across his lap they laid the cloven horn and the hilts and shards of his sword; beneath his feet they put the swords of his enemies.
The reason for this is obvious, as we learn in Aragorn's second verse of his departure song:
Beneath Amon Hen I heard his cry. There many foes he fought.
His cloven shield, his broken sword, they to the water brought.
Despite this, Faramir doesn't seem to have noticed it's absence when he came across Boromir's body (this is shown as if it were a dream-sequence in the Extended Edition of the Two Towers):
A broken sword was on his knee. I saw many wounds on him. It was Boromir, my brother, dead. I knew his gear, his sword, his beloved face. One thing only I missed: his horn. One thing only I knew not: a fair belt, as it were of linked golden leaves, about his waist.
We can therefore conclude (1) that Boromir's shield was destroyed during his fight with the Orcs, and (2) that he didn't leave Gondor with it but instead picked it up somewhere between his departure from Gondor and arrival in Rivendell, perhaps in Rohan.
By this reasoning the shield that Boromir arrived in Rivendell with was not a standard Gondorian shield, but Tolkien doesn't actually describe what these look like, so we can't say much on that count.
Tolkien does however describe what shields from Rohan look like, in his description of the arming of Aragorn and Legolas before Helm's Deep:
Now men came bearing raiment of war from the king's hoard and they arrayed Aragorn and Legolas in shining mail. Helms too they chose, and round shields: their bosses were overlaid with gold and set with gems, green and red and white.
Finally, adressing your question of "Is it common for a soldier to bear the same shield in all cases like a sword?", I would say "no". A shield is a defensive item, it gets damaged, and when it is too damaged to be useful any more it will be thrown away and replaced.