While there are certainly other "anointed ones" in the Old Testament, the Bible is abundantly clear that the Jews were waiting for the Anointed One, the Messiah.
King David was most definitely waiting for Him:
Psalm 2 (a Messianic psalm, meaning it refers to The Messiah, not many messiahs)
2 The kings of the earth take their stand, and the rulers conspire together against the Lord and His Anointed One:
3 “Let us tear off their chains and free ourselves from their restraints.”
David isn't speaking here about any ordinary "anointed one." He is talking about The Moshiach, Yeshua. Or in English, Jesus. In Arabic, Yesuwa. How do we know this?
Through the really big hint David gives us in the last two verses (italics mine):
11 Serve the Lord with reverential awe
and rejoice with trembling.
12 Pay homage to the Son or He will be angry
and you will perish in your rebellion,
for His anger may ignite at any moment.
All those who take refuge in Him are happy.
David's own son, Solomon, also wrote about this unique Son:
The Coming Messiah is prophesied about in at least 15 other Messianic psalms, all of them referring to a single Person, not multiple messiahs:
And here are some other examples from the prophets which show that the Jews were expecting just one Messiah:
The prophet Isaiah prophesied about one Messiah, for example in Isaiah 7, Isaiah 9, and Isaiah 53.
Jeremiah prophesied about one Messiah in Jeremiah 23:5.
Zechariah prophesied about one Messiah in Zechariah 3:8.
Micah prophesied about one Messiah in Micah 5:2-6.
Daniel prophesied about one Messiah in Daniel 9:25-26.
Those are a few examples from the Old Testament.
But what do we find the Jews doing in the New Testament?
They are still waiting for their One Messiah.
Upon their return to their homeland (which in itself was a fulfillment of Daniel's prophecies), the Jews were still looking toward the Coming of the Messiah as they were being oppressed by the Romans.
Mary, Elizabeth and Zechariah were waiting for Him. We read their stories in Luke 1.
Simeon and Anna were waiting for Him. Their stories are in Luke 2.
Jesus spoke about Who Messiah was to the unbelieving Jews, who understood completely Who it was He was referring to:
41 While the Pharisees were together, Jesus questioned them,
42 “What do you think about the Messiah? Whose Son is He?”
“David’s,” they told Him." Matthew 22
Notice that they didn't say, "There are many messiahs. Which one are you talking about?"
John the Baptist was waiting for Him. We read about that in John 1.
The disciples, who were simple fishermen and not rabbis or priests in the Temple, were also expecting the Messiah:
"He first found his own brother Simon and told him, “We have found the Messiah!” (which means 'Anointed One')." John 1:41
Even His Jewish mockers used the term together with a royal title, which gives us a hint at who they understood Messiah to be:
"Let the Messiah, the King of Israel, come down now from the cross, so that we may see and believe.” Even those who were crucified with Him were taunting Him." Mark 15:32
So it is obvious that the Bible disagrees with you that "messiah just means someone anointed by God." We need only to read the verses in their context to see that.
But what does the angel announce in Luke 2:11?
"Today a Savior, who is Messiah the Lord, was born for you in the city of David."
Jesus, the One Messiah, Messiah the Lord, had been born and the Jews could stop waiting.
Or at least, the ones who recognized Jesus for Who He is could.
For everyone else, even to those who reject Him as Messiah today, 2,000 years later, Jesus' words to the unbelieving Jews apply:
23 “You are from below,” He told them, “I am from above. You are of this world; I am not of this world.
24 Therefore I told you that you will die in your sins. For if you do not believe that I am He, you will die in your sins.” John 8