Let us touch particle base here, because you are talking of photons, and not of light beams.

There is a great difference between a light beam behavior, because it is a classically well modeled state, it has a stress energy tensor and would follow the geodesics of general relativity. As S McGrew states the attraction would be very very small because gravity is a weak force.

But when talking of photons we are no longer in the classical regime, we are in the quantum mechanical frame. Even ignoring gravity, photon photon interactions are very improbable, though they exist. Wikipedia has an article on this. This also makes light beams non interacting but passing through each other with just a superposition, the interactions would be happening by individual photons from each beam and would be of very low probability.

A Feynman diagram (box diagram) for photon–photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

This diagram is a recipe for calculating the integral which will calculate the probability of interaction between two photons due to the electromagnetic field. The probability is small because the vertices contribute as $(1/137)^2$ multiplicative to the integral.

Coming to gravity, we do not yet know that gravity is quantized, and has the equivalent particle to the photon, the graviton. It is effectively quantized and it is accepted that the graviton exists, but there is no final definitive theory for the quantization of gravity, proof of this is part of the current holy grail for theoretical physics.

If the graviton exists, the gravitational attraction at the photon level will be calculable by a similar diagram , where the contribution of gravity at each vertex would be exponentially small, as the gravitational constant is tens of orders of magnitude smaller than the electromagnetic constant which gives rise to the attraction at the particle level.

At best , if the two photons do scatter with a graviton intermediate, they will just change direction, but the huge probability is that they will not interact gravitationally.

How photons build up the classical beam, which is bent by large gravitational fields, is a story that needs a lot of mathematics to be understood. One has to be careful when talking of photons and light, photons are not bricks building up light, they are modeled by complicated complex functions which superposed build up the classical light wave.

Two photons cannot become "one photon" they will scatter through each other and move off conserving momentum and energy in the interaction. The scatter might be in the "attractive" direction , but there cannot be a "joining".

Let us touch particle base here, because you are talking of photons, and not of light beams.

There is a great difference between a light beam behavior, because it is a classically well modeled state, it has a stress energy tensor and would follow the geodesics of general relativity. As S McGrew states the attraction would be very very small because gravity is a weak force.

But when talking of photons we are no longer in the classical regime, we are in the quantum mechanical frame. Even ignoring gravity, photon photon interactions are very improbable, though they exist. Wikipedia has an article on this. This also makes light beams non interacting but passing through each other with just a superposition, the interactions would be happening by individual photons from each beam and would be of very low probability.

A Feynman diagram (box diagram) for photon–photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

This diagram is a recipe for calculating the integral which will calculate the probability of interaction between two photons due to the electromagnetic field. The probability is small because the vertices contribute as $(1/137)^2$ multiplicative to the integral.

Coming to gravity, we do not yet know that gravity is quantized, and has the equivalent particle to the photon, the graviton. It is effectively quantized and it is accepted that the graviton exists, but there is no final definitive theory for the quantization of gravity, proof of this is part of the current holy grail for theoretical physics.

If the graviton exists, the gravitational attraction at the photon level will be calculable by a similar diagram , where the contribution of gravity at each vertex would be exponentially small, as the gravitational constant is tens of orders of magnitude smaller than the electromagnetic constant which gives rise to the attraction at the particle level.

At best , if the two photons do scatter with a graviton intermediate, they will just change direction, but the huge probability is that they will not interact gravitationally.

How photons build up the classical beam, which is bent by large gravitational fields, is a story that needs a lot of mathematics to be understood. One has to be careful when talking of photons and light, photons are not bricks building up light, they are modeled by complicated complex functions which superposed build up the classical light wave.

Two photons cannot become "one photon" they will scatter through each other and move off conserving momentum and energy in the interaction. The scatter might be in the "attractive" direction , but there cannot be a "joining".

Photons always move with velocity c. There exists a center of mass system for your two photons, but they can never be one, because of energy and momentum conservation, a single photon has no center of mass due to its c velocity.

In view of above lets see your question:

would 2 photons on an uninterrupted and completely isolated path eventually gravitate towards each other and pass each other,

No. they can just scatter with a very small probability due to the gravitational exchange, the electromagnetic would dominate but it will still be very small.

then gravitate again in an oscillatory fashion, while the distance from an imaginary midpoint slowly decreases.

No, they scatter away with velocity of light c.

If this would happen, wouldn’t we be incapable of telling this combination of two photons apart from a single photon other than its increased energy than a normal photon and increased gravitational pull.

This cannot happen as said above.

Let us touch particle base here, because you are talking of photons, and not of light beams.

There is a great difference between a light beam behavior, because it is a classically well modeled state, it has a stress energy tensor and would follow the geodesics of general relativity. As S McGrew states the attraction would be very very small because gravity is a weak force.

But when talking of photons we are no longer in the classical regime, we are in the quantum mechanical frame. Even ignoring gravity, photon photon interactions are very improbable, though they exist. Wikipedia has an article on this. This also makes light beams non interacting but passing through each other with just a superposition, the interactions would be happening by individual photons from each beam and would be of very low probability.

A Feynman diagram (box diagram) for photon–photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

This diagram is a recipe for calculating the integral which will calculate the probability of interaction between two photons due to the electromagnetic field. The probability is small because the vertices contribute as $(1/137)^2$ multiplicative to the integral.

Coming to gravity, we do not yet know that gravity is quantized, and has the equivalent particle to the photon, the graviton. It is effectively quantized and it is accepted that the graviton exists, but there is no final definitive theory for the quantization of gravity, proof of this is part of the current holy grail for theoretical physics.

If the graviton exists, the gravitational attraction at the photon level will be calculable by a similar diagram , where the contribution of gravity at each vertex would be exponentially small, as the gravitational constant is tens of orders of magnitude smaller than the electromagnetic constant which gives rise to the attraction at the particle level.

At best , if the two photons do scatter with a graviton intermediate, they will just change direction, but the huge probability is that they will not interact gravitationally.

How photons build up the classical beam, which is bent by large gravitational fields, is a story that needs a lot of mathematics to be understood. One has to be careful when talking of photons and light, photons are not bricks building up light, they are modeled by complicated complex functions which superposed build up the classical light wave.

Let us touch particle base here, because you are talking of photons, and not of light beams.

There is a great difference between a light beam behavior, because it is a classically well modeled state, it has a stress energy tensor and would follow the geodesics of general relativity. As S McGrew states the attraction would be very very small because gravity is a weak force.

But when talking of photons we are no longer in the classical regime, we are in the quantum mechanical frame. Even ignoring gravity, photon photon interactions are very improbable, though they exist. Wikipedia has an article on this. This also makes light beams non interacting but passing through each other with just a superposition, the interactions would be happening by individual photons from each beam and would be of very low probability.

A Feynman diagram (box diagram) for photon–photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

This diagram is a recipe for calculating the integral which will calculate the probability of interaction between two photons due to the electromagnetic field. The probability is small because the vertices contribute as $(1/137)^2$ multiplicative to the integral.

Coming to gravity, we do not yet know that gravity is quantized, and has the equivalent particle to the photon, the graviton. It is effectively quantized and it is accepted that the graviton exists, but there is no final definitive theory for the quantization of gravity, proof of this is part of the current holy grail for theoretical physics.

If the graviton exists, the gravitational attraction at the photon level will be calculable by a similar diagram , where the contribution of gravity at each vertex would be exponentially small, as the gravitational constant is tens of orders of magnitude smaller than the electromagnetic constant which gives rise to the attraction at the particle level.

At best , if the two photons do scatter with a graviton intermediate, they will just change direction, but the huge probability is that they will not interact gravitationally.

How photons build up the classical beam, which is bent by large gravitational fields, is a story that needs a lot of mathematics to be understood. One has to be careful when talking of photons and light, photons are not bricks building up light, they are modeled by complicated complex functions which superposed build up the classical light wave.

Two photons cannot become "one photon" they will scatter through each other and move off conserving momentum and energy in the interaction. The scatter might be in the "attractive" direction , but there cannot be a "joining".