In North America, we sometimes encounter people who say that they do not recognize Islam as a religion. People hoping for some clarification on this matter from the field of religious studies will find only partial solutions as they discover that academic scholars do not all agree on a common definition of religion. Anyone offering a definition of religion is essentially stating who they are, hence there are multiple definitions of the term "religion."
To be very simple in explaining about Islam as a religion, there are some major salient features of Islam generally that may be recognizable and convincing to Christians and Jews. Christians and Jews will be familiar with and expect to see religious doctrine. Based in interpretations of the Quran and the example of the Prophet Muhammad, there is a substantive body of religious doctrine that has developed since the days of the pristine Islam of the Prophet Muhammad and His Companions up into the modern period. To make very brief selections of examples of this body of Islamic doctrine, Muslims talk about five pillars: the testimony of faith, that there is no God but Allah and Muhammad is his Messenger, almsgiving (Zakaat), prayer (Salaat, Dua), fasting (Sawm) and pilgrimage (Hajj). Notably, there is belief in divinely obedient angels as well as other created beings (Jinns), destiny, a day of judgment and an afterlife. There are different schools of interpretation that do function, sometimes more dynamically and sometimes very slowly, but these are not necessarily as rigid as some would have us believe.
Islam has a belief in one supreme God (Allah), who is the very same one true God described in the Ten Commandments. There is no "other" Creator of the universe and everything in it. Scripture is critically and centrally important in Islam as Muslims revere the Quran in its original Arabic as the uncorrupted Word of God. How this is properly interpreted is another matter but the Quran is recognized by all Muslims. The Quran mentions the Torah, Psalms and the Gospel, although the obligation to follow these is recognized as canceled by most Muslims. The important content of these is confirmed, sometimes restated, in the Quran. For example, the Ten Commandments are reiterated in different places in the Quran. Christians and Jews recognize prophets and Islam recognizes prophets who have borne revelation from God to humanity at different points in history. A number of these are mentioned in the Quran, including Adam, Noah, Abraham, Joseph, Moses, Solomon, David, Jesus and the Final Prophet, Muhammad (Peace be upon them all). Muhammad's superlative prophetic example is taken along with the Quran as authoritative for Muslims.
Muslims are immersed in prayer. There are obligatory prayers, five times per day: these are dawn, noon, afternoon, evening and nighttime prayers. There are voluntary prayers that can be done in addition to these obligatory prayers, in imitation of the Prophet Muhammad. These follow a set format, a "liturgy." There are supplications that can be done in addition to these prayers. Sufis will have whole litanies done in remembrance of God and other spiritual practices. Christians and Jews have fasting in their traditions and Muslims have a whole month of Ramadan dedicated to fasting. There are other fasts in addition.
Muslims practice almsgiving and charity in the form of Zakaat and also in the form of Khums and Sadaqa. Pilgrimage is an important part of the religion, especially the Hajj pilgrimage to Makkah (Mecca). There are other forms of pilgrimage in Islam where Shiites will visit shrines and mausoleums and Sufis will visit shrines of saints.
Holidays also exist in Islam. There are two major holidays that all Muslim are familiar with. There are other days that are celebrated traditionally. The birthday of Muhammad is often commemorated and there is a day to commemorate the Night Journey and Ascension of the Prophet Muhammad.
Christians will often criticize the seeming lack of emphasis on love in Islam, compared to its emphasis in Christianity. Indeed, we do need to speak more about love. The website of A Common Word has an interesting text for free download -- the book "Love in the Holy Qur'an" by Ghazi bin Muhammad (2010 Kazi Publications; also in Arabic). Then, of course, there is a whole body of Persian and Arabic love poetry.
We can wonder about anyone still doubting whether Islam is a religion after hearing these evidences. Different people have different motivations for making statements expressing doubt. While certain persons may have malignant motives, more ordinary and benign, well-intentioned types may be simply curious and might enjoy a discussion. For these nicer people, who will allow truth to unfold, we should kindly oblige them if they show an interest.
I suspect that people who have made such statements doubting that Islam is a religion often do have knowledge and a real conception of what a religion is and they are able to listen to a discourse if it is respectful. This description of salient features of Islam is enough to maintain that Islam certainly is a religion. If people are respectfully curious, we really should provide a respectful and sincere answer in return, which is best. Those are the better conversations and we should establish better interfaith conversations.