The word "son" has a variety of meanings depending upon the context. For example, Paul calls Timothy his son (1 Timothy 1:2, 18) and yet Timothy was not his offspring. In the Bible, Israel is called God's son (Exodus 4:22), angels are called sons of God (Job 1:6; 2:1; 38:7), and Christians are called sons of God (Romans 8:14,19; Gal 3:26). In each of these cases, the word "son" means something different.
In some cases, father/son refers to the loving and close relationship that exists between two people. In others, it refers to the authorial relationship where the father rules over the son. In other cases, it deals with the similarity of traits, characteristics, or behaviors. A child will often be "like" the parent. At other times, it refers to the relationship of property or rights. A son will inherit the father's position and property. In other cases, it is talking about the source or origin of one thing to another. So, for example, Jesus calls the Devil the father of lies (John 8:44) since he is the source of the first lie. Lastly, father/son refers to offspring in the sense of giving "birth".
So, when the Bible says that Jesus is the "only Son of God", what does it mean?
It is helpful to understand that the phrase "Son of God" when applied to Jesus is not just a description, it is also a title. For example, while there a many different presidents in the United States, there is only one President of the United States. When I talk about the President, you know who I am talking about because that is his title.
So, why does Jesus have that title? What makes him the "Son of God" and the only one? Well, to answer that question, we need to ask, "How did the people of Jesus' day understand it?"
When Jesus used this title to describe Himself, it made some people angry. In fact, they sought to kill him. They accused Him of blasphemy, a most horrendous sin, in which a person curses God. In John 5:18, the reasons behind this are explained more clearly. It says, "the Jews were seeking all the more to kill Him, because He not only was breaking the Sabbath, but also was calling God His own Father, making Himself equal with God." (NAS) By claiming to be the "Son of God", Jesus was claiming to be equal with God. That is how they Jewish people understood the title of "Son of God". This is very different than how we use the phrase in our modern day. The word "son" in English is never used in this way which is why people don't understand what the title "Son of God" actually means. It means that Jesus is God.
That is why the Jews accused Jesus of blasphemy because they did not believe He was in fact God. They were accusing Jesus of lying. At his trial, the high priest questioned Jesus saying, "Are You the Christ, the Son of the Blessed One?" Jesus responded by saying "I am ; and you shall see THE SON OF MAN SITTING AT THE RIGHT HAND OF POWER, and COMING WITH THE CLOUDS OF HEAVEN." (Mark 14:62-63) They then accused Him of blasphemy which according to Jewish law is punishable by death (see Leviticus 24:16).
As for your question as to whether any other parts of the Bible show that Jesus is the only Son of God, the answer is "yes". Everywhere it talks about Jesus' divinity, every time it calls Jesus "Lord", and every place it calls Jesus "Christ" are all references to Him being the "Son of God". Those three titles, "Lord", "Christ", and "Son of God" are all talking about Jesus position and relationship over and to humanity.