Once when I was going door to door to present the message of the Good News, I met a person who would not change his 'religion' for another one. I asked him how he became religious, and he replied it is by child baptism. I politely pointed out to him that a personal relationship with the Lord is needed to be religious (Ephesians 2:8). But he held on to his belief that his baptism to the Catholic organization paved his way to be religious.
James 1: 27 says, "Religion which is pure and undefiled in the sight of God, even the Father, is this: to look after orphans and widows in their affliction with a view to ascertaining their needs and supplying them, and to be keeping one's self unspotted from the world." In this context, the word 'religion' means a Christian service being done or performed.
So why do many Christians say they are not religious? Why do they say they don't belong to a religion? It is not what the New Covenant teaches. The word 'religion', in the New Covenant Greek θρησκεία, means 'devout' and 'pious'.
Now, for the word 'irreligious', look at 1 Timothy 1:9, "...for those who are destitute of reverential awe towards God and for sinners, for unholy ones and for those who are irreligious..." This verse teaches us the opposite of a religious person. James 1:26 and 1 Timothy 1:9 shows us the contrast of what it is to be religious and irreligious as described in the New Covenant.
How do we become religious then? First, to be saved is a must to become religious (Ephesians 2:8). Then by the Holy Spirit who lives in us, at our conversion, we can serve religiously (Ephesians 2:9).
Forget about how the world uses biblical terms and don't let the world dictate you how to talk. But be faithful to the word of God; that's all God the Father is asking from us. The language of the New Covenant should remain pure from any encumbrance (interference/hindrance). Don't be afraid of saying, "I am religious" or "I belong to a religion", because, if you are following what James 1:27 says, you are religious and you belong to a religion-- the religion founded by the Lord Jesus Christ.