Sikhism began as an attempt to unify Hinduism and Islam. What makes this attempt inherently problematic? Is it more productive or counterproductive to seek unity between religions by inventing a third, different religion?
I think that the initial presupposition within the question is an interesting one. There is both a desire to unify spiritual beliefs that are posited as being opposite. There is also an intrinsic challenge in doing so. The initial presentation of Sikhism to unify Hinduism and Islam is problematic because both spiritual notions of the good start out from opposite points of reference.
In Hinduism, the fundamental starting point is the belief in Atman, defined as "the true self of an individual beyond identification with phenomena, the essence of an individual." Atman is presented as a reality of consciousness. It is not associated with a particular individual of worship. For example, one cannot pray to Atman. Rather, it is the reality of individual consciousness. The fundamental starting point in Islam is different, as it predicates that the teachings of Muhammad as where the individual must focus their energies. The teachings of Muhammad the prophet is the word of Allah, and submission to this reality defines the individual and their religious expression. Both starting points are difficult to reconcile because they begin in two different realms. Muhammad and Allah are directly identified in Islam, while Atman is a universal essence that becomes the starting point for the Hindu. This condition makes the reconciliation of both fundamentally difficult or problematic.
It is a matter of individual perception in arguing whether or not this is productive or counterproductive. I think that in looking at Sikhism, one recognizes that its foundational intent is productive. There is a universality of the divine in Sikhism, something that represents an attempt to bring together human beings: "The central teaching in Sikhism is the belief in the concept of the oneness of God." Guru Nayak argues that "Realization of Truth is higher than all else. Higher still is truthful living." He also suggests that such a realization underscores all religious worship: "When He so willed, He created the world. Without any supporting power, He sustained the universe. He created Brahma, Vishnu and Shiva; He fostered enticement and attachment to Maya." Sikhism argues that this universal "oneness" of the divine creates all religious worship. It represents an attempt to resolve religious differences through the establishment of a universal condition of being.
The universality of intent collides with the reality of human expression. The religion's intent faces challenges when its reality ends up becoming its own singular notion of the good. There is much in way of laudable intent in terms of religious experiences being developed in order to bring about unity and convergence. If that becomes its own singular construction of the good, it becomes a regressive cycle. Human beings have shown a desire to place religious expressions in the camp of singularity and exclusivity when the intent of the spiritual experience has been conceived quite differently. However, the intent must be examined, and anytime that acceptance and convergence is sought, that cannot be a bad thing.
Yes, Sikhism began with an attempt to bring unification in Hinduism and Islam. The main motive was to bring faith each other's community. The Islamist were ruling India at that time period and wanted to spread their religion. Hindu kingdoms were all spread and fighting among themselves. This way they could not unite and even lost some battles from Islamist. Therefore distrust prevailed. Until Sikhism came in with the preaching of the holy preachers who taught to unite and respect each other freedom, integrity and religion.
It is not quite correct to say that Sikhism began as an attempt to unify Hinduism and Islam. Yes, it is true that Guru Nanak - the great saint, who was the first and the most influential of the ten Gurus (religious teacher) responsible for shaping the religious tenets of Sikhism - did not discriminate between people on the basis of their religion. Yet the religious concepts professed by him were based on the ancient Hindu philosophy, that is a major pat of Hindu Religious belief today and was at the time of Guru Nanak also. We can say that non discrimination between people based on religion or other criteria like race, or social status is also a part of such Hindu Religious.
As I see it, Guru Nanak did not try to start or develop any new religion. He only tried to promote good human values which happened to be rooted in core Hindu religious beliefs. In doing so he borrowed from the works of many well known Hindu saints. Some of these saints like "Kabir" were explicitly critical of bad practices prevalent in both Hinduism and Islam. But, by and large, teachings of these saints, including that of Kabir, and of Guru Nanak were based on typical Hindu Beliefs.
Also it is important to note that like original inspiration behind many of the great religions, Guru Nanak did not try to start a new religion. He was only trying to inculcate good human values among all people. The listened to him and accepted his ideas were mostly Hindus, but included some Mohammedans also. These people started calling themselves "sikh", implying the ones who have "learned" from Guru Nanak. It is interesting to that the distinction between Hindus and Mohammedans among these so called Sikhs always remained.
Later as Sikh community became bigger and stronger under the leadership of subsequent religious Gurus, it faced oppression from the Mohammedan rulers in India. In response it got transformed into a warrior community fighting against Mohammedan oppression. Thus, Sikhism came in direct confrontation with Islam. In these circumstances, Sikhism gradually developed in to a separate formal religion, rather than being just a group of people following the teachings of Guru Nanak, while continuing to maintain their other religious beliefs.
In conclusion, I will say that, though Sikhism as a concepts may be considered as an attempt to unify Hinduism and Islam, but as a formal religion it has been in direct confrontation with Islam.
Next coming to the second part of the question, that is,
"Is it more productive or counterproductive to seek unity between religions by inventing a third, different religion?
The biggest flaw in trying to invent a new religion to to seek unity between religions is the need to accept beforehand that "none of the existing religions are good enough". This hurts the ego of the people. Generally we do not use the word "ego', Instead we say "religious sentiments". The best approach for every well meaning person is to use discrimination in accepting what is good in his own religion and discard what is best.
More capable and public spirited people can try to reform their own religion rather than start a new. If such attempts result in formation of a new religion, we can not help it. But that is not as bad as starting out by demolishing an existing religion. In this context, it is worthwhile noting that Jesus Christ was perhaps trying to reform Judaism, rather than establish a new religion called Christianity.