No, it wasn't. Although there's a persistent myth that connects it to the 18th century erotic novel Fanny Hill, the first known attestations of the term in a vulgar sense are from some music hall songs from the late 1830s, several decades after the publication of Mansfield Park. Although even a half-century later it's still apparently considered just an innocent name even in pornographic contexts, as Spedding and Lambert argue:

However, although fanny had become a slang term for vagina by the late 1800s, not every reference to Fanny (the name), even in this period, involves a sexual pun. In Sellon’s The Ups and Downs of Life —a book “rich in slang” according to Clifford J. Scheiner — the name Fanny appears without it carrying any obscene meaning. Likewise, in The Pretty Women of Paris (1883) — a pornographic text that uses a wide variety of sexual slang— the name Fanny is used numerous times in the description of upper-class French prostitutes, again, without its carrying any obscene meaning. Clearly, if pornographers could still use the name without any sexual connotation in the late-nineteenth century, it is likely that many conventional writers in this period were completely unaware that fanny carried a sexual meaning.

Fanny did remain a popular name well into the 20th century. If you look at those many websites that have name statistics culled from census data it seems it peaked around 1920 in the USA before dropping off to virtually nothing around 1940. In the UK, according to the statistics this site compiled, it was the 69th most popular name for newborn girls in 1900.

It's safe to say it was more popular a century ago than it is now in the UK as well.

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Patrick Spedding, & James Lambert. (2011). Fanny Hill, Lord Fanny, and the Myth of Metonymy , Studies in Philology, 108(1), 108-132.