Avraham’s camels were immediately recognizable.
In Parashat Chayei Sara, we find an iconic comment by RaShI, based upon Rabbinic tradition mentioned in Beraishit Rabba 59:11 :
Beraishit 24:10
And the servant took ten camels, of the camels of his master (this phrase appears superfluous—would Eliezer take camels belonging to others?), and departed; having all goodly things of his master’s in his hand; and he arose, and went to Aram-Naharaim, unto the city of Nachor.
RaShI s.v. MiGemalai Adonav
They were distinguishable from other camels, because they would go out muzzled due to a concern regarding thievery, in order that they would not graze in the fields belonging to others.
(An explanation of a detail in v. 32 further supports this idea:
Ibid. 32
And the man came into the house, and he ungirded the camels; and he gave straw and provender for the camels, and water to wash his feet and the feet of the men that were with him.
RaShI s.v. VaYifatach
He removed their muzzles, for he had sealed their mouths in order that they not graze along the way in fields belonging to others.)