Does Islam claim to have a unique claim to the truthful religion? Is there a difference of opinion within Islam on the answer to this question?
No there is no difference in opinion, Islam is the only true religion, the religion of God that He sent down to all the Prophet's (peace be upon them).
إِنَّ الدِّينَ عِندَ اللَّهِ الْإِسْلَامُ ۗ وَمَا اخْتَلَفَ الَّذِينَ أُوتُوا الْكِتَابَ إِلَّا مِن بَعْدِ مَا جَاءَهُمُ الْعِلْمُ بَغْيًا بَيْنَهُمْ ۗ وَمَن يَكْفُرْ بِآيَاتِ اللَّهِ فَإِنَّ اللَّهَ سَرِيعُ الْحِسَابِ
Indeed, the religion in the sight of Allah is Islam. And those who were given the Scripture did not differ except after knowledge had come to them - out of jealous animosity between themselves. And whoever disbelieves in the verses of Allah , then indeed, Allah is swift in [taking] account.
Surat Al Imran Ayah 19
Allah Also said:
وَمَن يَبْتَغِ غَيْرَ الْإِسْلَامِ دِينًا فَلَن يُقْبَلَ مِنْهُ وَهُوَ فِي الْآخِرَةِ مِنَ الْخَاسِرِينَ
And whoever desires other than Islam as religion - never will it be accepted from him, and he, in the Hereafter, will be among the losers.
Surat Al Imran Ayah 85
Furthermore, unlike the other religions, Islam brings clear solid proof to backup/support this "claim (as you say it)".
This a complex topic. Islam is not only the religion of Mohammad pbuh it is the pure faith in God and thus all God's messengers are muslims although the name is thought to have been first used by Abraham (according to muslim tradition). I will leave it to other members who are more knowledgeable to give an extensive answer.
I want to point to this verse in Quran (Al Baqara verse 62):
[002:062] Those who believe (in the Quran), and those who follow the Jewish (scriptures), and the Christians and the Sabians,- any who believe in God and the Last Day, and work righteousness, shall have their reward with their Lord; on them shall be no fear, nor shall they grieve.
Arabic: 2:62 ان الذين امنوا والذين هادوا والنصارى والصابئين من امن بالله واليوم الاخر وعمل صالحا فلهم اجرهم عند ربهم ولاخوف عليهم ولاهم يحزنون
some exegists say that the verse has to do with those who followed aforementioned creeds before the seal of prophets Mohammad came with God's last direct message to humanity others say it still applies to all of those who have a pure faith in God.
God knows best