How do Christians who emphasize the "religion vs. relationship" dichotomy respond to claims of "relationship with God" in other religions?

A basic premise of Christianity is that an evil one tempted through deception; the first woman fell for it, her husband chose to go along with her, and thus the evil of sin and deception began to grow and spread, like a cancer, as more and more people were born and found themselves imperfect sinners who could not relate to God without him taking the initiative. Which he did. I mention this because there IS a way for sinners to relate to God. The trouble is, the deceiver has never stopped working hard at fooling people into supposing they can relate to God in ways other than the one way God has stipulated. People keep falling for Satan's temptation to "be like God, knowing good and evil" by partaking of that which God has forbidden. One 'bite' of the symbolic 'fruit' of Satan, and the cancer of his deceptions spreads in the individual.

This has resulted in myriad religions, with only one being recognised by God as the acceptable way of approach to him - via the covenant he established with those who had faith in him (Noah, Abraham, Jacob who fathered the nation of Israel at the time of Moses.) None of the other world religions back then could bring anybody into relationship with God, but individuals could turn and be joined through the covenant. Then came the era of Jesus Christ and his ministry on earth, where he founded his church which the gates of hell would never overcome (Matthew 16:11-19). Jesus stated that HE would build his church, with himself as the one and only mediator between God and man. He stated that nobody could come to God the Father except through him. It is faith in what Jesus accomplished on the cross (because of who he is) that brings anybody into relationship with God.

All attempts at all religions that do not put total faith in Jesus Christ as the resurrected Son of God are doomed to failure because they have fallen for satanic deceptions. They are trying to appease God by what they do; to earn God's approval by religiosity. Take, for example three religions already mentioned in answers and comments. The New Age for one. Sure, they enthuse about 'spirit guides', wonderful experiences (visions, even). But they could hardly care less which deity you like (you could even be an atheist - makes no difference to the experiences you could have). You can acclaim a hippy Jesus, a Buddhist Jesus, or no Jesus at all. (Oh, and Buddhists don't need to believe in any god because their goal is to 'merge' with the impersonal - not relate to a personal deity.) Anything goes in the New Age but not according to what the Father and the Son have already communicated to mankind. The New Age ignores that because they promote a pick-and-mix system of religiosity. The supposed 'relationship' New Agers claim to have with whatever deity they uphold is more likely linked to hallucinatory drugs than anything actually spiritual, or, if it is, then the spirits involved are agents of the deceiver. I've read Shirley McLaine's books on the New Age, and personally know a New Ager who did personal horoscopes for people, until he realised he was into the occult and was converted to faith in Christ. The covenant in the Hebrew Scriptures shows that any alliance with the occult brings God's adverse judgment. Nobody can relate to God whilst practicing the occult.

Second, take Islam. If people think Muslims claim to have a personal relationship with Allah, they need to think again. I quote from objective sources:

"The Arabic word islam means 'surrender', and the use of the term in various passages of the Qur'an shows that this is surrender to God. Thus Islam is the religion of surrender to God, and one might say that among the great religions Islam is characterized by its emphasis on the omnipotence and transcendence of God and humanity's servile or slave status before him... The teaching of Muhammad was a result of divine revelation. All his teaching is believed to be recorded in the Qur'an and is defined as surrender to the will of God as so written. In addition [to the 5 Pillars of Islam] 5 observances are expected of every Muslim. By these observances every believer hopes to earn salvation at the Day of Judgement." (The Encyclopedia of World Faiths pp154 & 165, Darton & Todd, published by Macdonald Orbis in 1987) [Emphases mine]

Because of beliefs about the Day of Judgment, it is not possible for individual Muslims to suppose that (prior to that Day) they can enjoy a personal relationship with God. They think that select human beings have been charged by God, and enabled by a spirit from the amr [i.e. God's will, command] to be his prophets, Muhammad being the 'seal' of the prophets. The 1st Pillar is to recite the Shahada about Allah being the only God, and Muhammad as his prophet. No other prophet occupies that elevated, unique position, so that Muslims would never dream of trying to approach God via Jesus Christ. Also, no Muslim can know until the Day of Judgment whether they have been submissive enough to God, and have done enough good deeds to outweigh their bad deeds, and so tip the divine scales of justice in their favour. Whilst they strive to diligently uphold Qur'anic moral law and order in fear of not meeting God's just laws, they hope that trust in God's mercy will also count as merit for them. Indeed, the Qur'an condemns lack of hope in God's mercy as a cardinal sin. This all adds up to never knowing the mind of God regarding their personal relationship to him until that fateful Day of Judgment. Another quote:

"The term 'fatherhood', in reference to God, is abhorrent to Muslims, because it denotes a physical relationship. To consider God as 'Father' is, for Muslims, to imply that God has a wife and an issue - both of which are blasphemous concepts. Nor are Muslims prepared to rationalize the term 'Father' in a metaphysical sense as 'Father of all mankind', because in Islamic teaching, human beings are chattels - creatures and servants of God, not His children. Nor again do Muslims recognize the idea of God the 'Father' as presented in the Christian concept of the Trinity. Trinitarianism is flatly condemned in Islam". (Islam, The Way of Submission by Solomon Nigosian, p189 & p117, Crucible 1987) [Emphases mine]

Third, take Hinduism. Some would claim that the Hindu Trimutri equates with the Christian Trinity doctrine in terms of being able to relate personally to the one God. Hinduism has a triad of deities, but it does not have a trinity. The Hindu triad requires three gods and is called 'Trimurti' and is worshipped in the form of the guru principle Dattatrya. Let me quote from a Hindu scholar on this:

"Sometimes the three gods Brahma (a personification of the impersonal Brahman), Vishnu and Shiva are grouped together in a triad called Trimurti, occasionally although wrongly described as a 'Hindu trinity'. In this representation of 'One God in three forms' Brahma is the creator, Vishnu is the preserver and Shiva is the destroyer. Well known from Indian sculpture, the triad has largely remained artificial and is unimportant to Hindu worship, which is much more strongly founded on the notion of Istadevata, the idea that an individual or a family chooses a specific god as the main focus for devotion." (Encyclopedia of World Faiths, Bishop & Darton, p 193 published by Macdonald Orbis 1987) [Emphasis mine]

Thus, when you suppose Muslims claim to have a personal relationship with Allah, that's false because that would be blasphemy for any Muslim. A Hindu cannot claim to have a personal relationship with Brahman because the three gods Brahma is merely a personification of the impersonal Brahman. New Agers cannot claim to have a personal relationship with an impersonal Universe, and their spirit guides are not God. That is my response, as a trinitarian Christian, to claims of other religions having a "relationship with God".