I'm not aware of any solid connection; There is slight speculation, but it's not very well linked and not a mainstream theory, and the relations are most likely just parallels. As you'll see below, even the splits within Hinduism can be viewed as parallels to splits within other western religions, like divides within early Protestantism. In fact, there was the same question as to whether salvation was predestined or a product of one's works. Those who argued for salvation being a product of one's works in the northern school said that it was much like a monkey who carries her offspring to safety, swinging, as her child clings on as best it can. in the southern school, salvation came like a cat picking up her kittens by the scruff of the neck and carries them about, without any effort.

As some history about Madhva's ideas...

This is a difficult subject to get into without really tracing all of the roots and branches, starting as early as Jainism. I'm going to be using lots of terms and names to talk about what exactly was going on at the time that may be unfamiliar, but I hope you look them up if you're interested and investigate a little more yourself. My main point will be that there was a lot of thinking and change going on in many Hindu schools (and Buddhist schools, Confucian, Daoist, etc.) and in order to really comprehend what you end up with, you have to look at the path that you took to get there. Most Eastern philosophy builds upon the past, rather than replace it.

But to throw some names out, three figures that shaped the vedanta schools of Hinduism (referring to the end of the Vedas, the Upanishads) are Shankara, Ramanuja, and Madhva. Each argued for a different relationship between Atman, the self, and Brahman, that notoriously hard to define concept that some refer to as something like "the real reality" as opposed to the illusion of permanence in this world.

Shankara is around in the 8th and 9th centuries. Born to the Brahmin caste (not to be confused with Brahman), Shankara lead an ascetic life starting at a very early age and is also responsible for founding four famous "monasteries" (mathas) in India, one in each cardinal direction (north, south, east, west). Most of his writings were commentaries on the texts that the vedanta schools focus on, the Upanishads, the Bhagavad Gita, and the Brahma Sutras (one of many Sutras, such as the Lotus Sutra, or the Kama Sutra). There are three points that Shankara outlines all his thinking with, underlining his belief in non dualism (that there is only one reality and that everything is a part of it): Only Brahman is real, the world is an illusion, and the self is part of Brahman. So, all change in this world is an illusion/distortion, and there is no creation, everything is just part of Brahman. There's this kind of partial reality vs ultimate reality position that is reflected in many points. For example, as for the ultimate reality of Brahman argued by Shankara, it is "God" without quality or attributes, just pure existence and consciousness and beyond any names or forms. Additionally, Atman (the "soul") is all part of Brahman, and it is only the distorted reality that keeps them separate. To break the cycle of reincarnation, samsara, Shankara puts forth that a kind of "self correcting" and logical thinking and wisdom are aids towards enlightenment, but only partial (the analogy is "this way is like a knife that can cut everything around it, but cannot cut itself"). It is only through "non thinking" that this can be achieved, with the help of aided study. A great and very famous analogy is that it is like 10 young men who make a river crossing. After crossing, one of them counts the group, and only counts 9. Another does the same, and also only counts 9. A passerby sees what's going on, and notices that each had counted all but himself, and points to each, saying "you are the tenth one". And thus, they realized that they were that, and always were.

On to Ramanuja, coming around the 11th or 12th century (there's some debate). He also became a Vaishnavite monk, and wrote commentaries on the same works as Shankara. But the conclusions drawn were not the same. Ramanuja felt that the Brahman defined by Shankara was too remote and abstract. Basically, it was very hard to incorporate temple Bhakti to Shankara's idea of Brahman. Ramanuja places vast importance on bhakti with an individual god, and also argues for qualified nondualisim. In Ramanuja's argument, Atman and Brahman are the same substance, but are of different qualities. I hesitate to make this comparison, but it's a bit like the Nicene Creed, in that it resolves apparent contradictions. So there's one reality of Brahman, but there are distinct parts and does change, just within Brahman. Additionally, the world is not an illusion, it is a reality, and souls exist and are distinct, but are not entirely independent. You can distinguish between the "you" and "that", but they're both ultimately the same. It might be a little confusing, so don't worry. In Ramanuja's idea of samsara, again, the escaping of the reincarnation cycle, instead of "returning" as part of Brahman, your soul escapes, and basks within the presence of Brahman. Another difference between Shankara and Ramanuja, is that while Shankara says the main problem is ignorance, and the solution is learning, Ramanuja argues that the main problem is unbelief, and the solution is bhakti. This emphasis on bhakti as being the way of salvation/enlightenment spawns a great amount of poetry as a form of devotion/bhakti and a lot of it is written as love poetry (think the Song of Solomon). Finally, caste, merit, and even religion don't impact one's reunion with Brahman.

Finally, now we come to Madhva in the late 12th, early 13th century and was part of the Brahmin caste, same as Shankara, and wrote commentaries on the same vedanta texts. Madhva is not a non dualist. Instead, Madhva is solidly a dualist and is the founder of dualistic vedanta. Madhva argued that there were two things, separate: Brahman and Atman. Additionally, he put forth that Brahman, matter, and self all are distinct and permanent, but matter and self are subordinate and dependent on the self-dependant Brahman. In Madhva's school, the focus is on the differences between Brahman and self, rather than the unity between the two. Brahman is different than self, Brahman is different than matter, matter is different than self, and self is different from self. Self takes on several different qualities, some free of samsara, some passing through samsara, and others will eternally continue through samsara (this last one is a completely new idea). To break out of samsara, one participates in scripture study and gains knowledge, and then Brahman lifts the veil, as a sort of salvation half by works, half by bestowment. Madhva's thinking also opens up much room for theism, as the Brahman described by Madhva has very clear characteristics, as opposed to, say, Shankara's idea of Brahman.

To reiterate, why did I go through all that stuff on Shankara and Ramanuja? Simply because it's important to see what the evolution of the vedanta schools was: from non dualism, to qualified non dualism, and then to dualism. Each of these three schools builds upon previous ideas and texts, and has their own influence, and reacts to the other vedanta schools. Shankara reacts to Buddhism, Ramanuja to Shankara, and Madhva to both. These ideas did not come about in a vacuum, and are very much a product of south and east asian thinking and movements.

If you'd like to delve more into the history of Hinduism and more, much of this information can be found in the Great Minds of the Eastern Intellectual Tradition from the Great Courses series.

Can you elaborate on where you saw similarities? If I had to guess, I'm assuming that most of these are going to center around the sort of "cosmic" Christology that had developed in certain circles of early Christianity (ultimately stemming from the Gospel of John, Philippians, Colossians, etc.).

There may be some slight connections that could be drawn based on, say, the shared Indo-European heritage of Indian and Greek thought (in a very general sense)...but most other things are just going to be slight typological similarities in particular aspects.