Robert Browning is naturally considered a Victorian poet, considering that he wrote during the time period of Victorian England. And yet Browning's work is simultaneously a revolt against some of the most well-defined aspects of that time, and a reflection of its characteristics.
Victorian England, named after Queen Victoria who was crowned in 1837, is marked by several social qualities: repressed sexuality, strict morality, an expansion of English imperialism, a focus on human inventiveness, and nascent doubt over man's place in the universe. With the world changing so quickly over the roughly 70 year-period, artists, scholars and scientists created and wrote from a place of unrest. Where perhaps most of them came down strong on one side of the period's many questions, Browning embraced the uncertainty of his time as a facet of human nature and psychology, and his poetry reflects not strong opinions but rather our tendency to waver between opposing views.
Perhaps the most well-known aspect of Victorian England was its 'prudish' attitudes on sex. Operating under the belief that women were not to be consumed with sexual lust, laws and social strictures forced men and women into entirely separate spheres. The hope was that secure, happy families could be created and by default a moral society. Browning's work takes great issue with such repression. Though he is by not means a libertine, he reflects in many poems the cost of such repression as an equally vicious reaction. Poems like "Porphyria's Lover" or "Evelyn Hope" show the grotesque side of such assumptions. Further, the class element of this Victorian idea (that women should prepare a nice home for a man's success) is shown to be equally vicious in poems like "My Last Duchess" and "The Laboratory."
Though Browning was not explicitly a political poet, his work does reflect doubts in the supremacy of England as Victorianism saw it. Consider poems like "Caliban upon Setebos," which proffer the thesis that we are all of us flawed creatures who know nothing of anyone save ourselves. The argument implicitly counters the Social Darwinist ideas that justified England's extreme imperialism.
Browning's time also saw great advances in human knowledge, but ones that came at the cost of a long-held Christian faith in the divinity of man. The Industrial Revolution opened up man's ability to exploit nature for his own gain, while new opportunities for education created new readers and thinkers, and new scientific discoveries - primarily Darwin's theory of evolution - led many to doubt that man was in fact a reflection of a supreme deity. While these advancements certainly improved quality of life, they also brought with them an age of doubt. Many writers embraced such a worldview and sought to express new ideas in the possibilities, but Browning explored both sides, questioning the value of a life without faith while also celebrating the possibilities of a man less tied to God. Poems like "Caliban upon Setebos" or "Rabbi Ben Ezra" confront these questions directly, but many others - like "Andrea del Sarto" - reflect a sophisticated concept of human psychology, one that suggests we are limited to our perceptions and entirely conditioned by the circumstances of our lives. These days not a radical idea, in Victorian England it was far more groundbreaking to suggest that there is nothing about us that is a priori divine and perfect, but instead that we each of us develop our own moral sense, and moreover have the ability to rationalize our moral sense as acceptable. Browning's love of drama was fed by such a worldview, since he was able to empathize with the perspectives of characters who otherwise preach attitudes we might find abhorrent. Browning was much enamored of the complications and potentials of human beings, and found great conflict in the way these elements tried to fit in with a bigger world.
The Victorian period followed directly what is known as the "Romantic period," during which poets explored the concepts of individuality as a key to transcendence. Browning, as a great admirer of the movement's best writers – Shelley and Coleridge amongst them - certainly never went full-fledged into Romanticism, but did recognize the power of hope and beauty that comes from self-knowledge and self-exploration. As such, he did not entirely accept that these doubts led to pessimism, though he did empathize with such pessimism, as seen in "Caliban upon Setebos."
All in all, Browning was a man of his time, both in the way he reflected the new Victorian learning and questioned some its assumptions on morality and behavior.