Did Brahms really leave Germany because he was dissatisfied with the political realities of that time?

SHORT ANSWER

Brahms most likely based himself in Vienna because of better opportunities there. The political situation in 'Germany' (Hamburg) does not appear to have had any relevance.

DETAILS

There does not seem to be any evidence that Brahms left Germany (Hamburg, to be specific) due to the political situation. In fact, 1862 was when Bismarck was appointed Minister President of Prussia, and Brahms was a great admirer of Bismarck:

He had a passionate admiration for Bismarck, was pleased to receive presents of each of his portraits, loved his speeches, and was familiar with everything that had been written about the Iron Chancellor. Three weeks before his demise, when the treacherous illness had robbed him of all pleasure in living, he complained to his friend Herr Arthur Faber that he could no longer retain what he read. “I only want to read about Bismarck; send me the book by Busch, Bismarck and His Men.”

Source: Heather Platt, Johnannes Brahms: a Guide to Research

Brahms greatly admired Bismarck, and knew many of his speeches and much of his writing by heart.

Source: Walter Frisch & Kevin C. Karnes (eds), Brahms and His World

Brahms was also a German nationalist.

Statements from members of Brahms’s circle, including Heuberger and Kalbeck, strongly suggest that Brahms was a German nationalist

Source: Walter Frisch & Kevin C. Karnes (eds)

Brahms had become a good Austrian and at the same time remained a faithful Reichs-German. He read the historical works of Sybel and Treitschke, and finally Oncken’s book about Kaiser Wilhelm, with the warmest sympathy and interest.

Source: Heather Platt

So why did Brahms move to Vienna? The excerpts below give a fairly clear picture. First, this from Britannica:

By 1861 he was back in Hamburg, and in the following year he made his first visit to Vienna, with some success. Having failed to secure the post of conductor of the Hamburg Philharmonic concerts, he settled in Vienna in 1863, assuming direction of the Singakademie, a fine choral society.

This from Gramophone: In Hamburg,

..he had written his magnificent First Piano Concerto. It was hissed at its premiere in 1859 and the year later he had a further disappointment when he was turned down for the conductorship of the Hamburg Philharmonic. Then an invitation to conduct in Vienna, his mother’s death in 1865 and father’s remarriage, loosened his ties with Hamburg. Eventually, in 1872, he decided to make the Austrian capital his base.

This from Music Academy Online:

Brahms also met with some success and some new friends in Vienna where he spent the winter of 1862 to 1863. While he was there, Brahms was bitterly disappointed to discover that the much-coveted position at the Hamburg Philharmonie had escaped him. In the spring of 1863, his disappointment was mitigated by an offer to conduct the Vienna Singakademie.

The Platt and Frisch & Karnes books state similar reasons to the above, with the former emphasizing that Vienna offered "numerous opportunities", while the latter quote a possible additional reason cited in Max Kalbeck’s book Johannes Brahms concerning the prima donna Louise Dustmann. Although the authors criticise Kalbeck for failing to provide sources, there is no doubt that Brahms had a close, maybe intimate, relationship with her and that this may have a been a factor, albeit a minor one.

Kalbeck claims that Dustmann was one of the allurements that brought Brahms to Vienna in 1862, having fallen under her spell at that summer’s Festival of the Lower Rhine. He says that she captivated him “by her voice and abilities” but neglects to say what her abilities were. One would like to know Kalbeck’s source. He writes that Brahms was in danger of losing his heart to her, and that once he reached Vienna Luise’s behavior toward him was even more seductive than at the Festival. “Certain it is that Brahms had an intimate friendship with her,” Kalbeck writes, “and that until his end (Dustmann outlived him by two years), he remained her faithful friend.” She performed in Brahms’s concerts, supplied him with opera tickets, studied his lieder with him, sang the premieres of over a dozen of his songs (including the famous Lullaby and the Liebeslieder Waltzes), and regularly received complimentary copies of his vocal music as they were published. Brahms was a frequent visitor to her apartment, which was within “astonishing proximity” to his own. The two summered in the same obscure retreats at the same time, Tutzing and Portschach.