Slavic identity seems to much stronger than something like Germanic identity or Romance identity or another similarly large group of related languages.
This question has many different facets that need to be considered.
Well... Yes and no. You'll find that people from different Slavic subgroups kinda have the same identity (ex. Russians, Ukrainians, Belarussians ; or the Yugoslav peoples ; or the Polish, the Silesians and the Tcheques ; for example), but if you take them all together, it really doesn't work (see Polish vs. Russians for example, they 'hate' each other more than the Germans and the French hate each other.)
It is often thought to have originated from a stream of thought called 'panslavism', a philosophy which was later espoused by the Czars around the second half of the 19th century, and used as a justification for expansion into Central Europe.
The most known consequence is probably the 1st world war. Russia's alliance with Serbia was pushed by the panslavic movement, and thereby partially caused WWI.
Some reading on this subject :
http://en.wikipedia.org/wiki/Pan-Slavism
Nicholas Valentine Riasanovsky, A History of Russia, p. 450, Oxford University Press US, 2006, ISBN 0-19-512179-1.
A. Grigorieva Pan-Slavism in Central and Southeastern Europe // Journal of Siberian Federal University. Humanities & Social Sciences 1 (2010 3) 13-21. [2]
This question is a little peculiar, because it's very difficult to measure "a sense of identity" in a meaningful way. While Pan Slavism is (or was) definitely a thing, non-russian Slavic ethnicities like Poles and Ukranians would definitely dispute any assertion that they are Slavic before they are Polish or Ukrainian, and there's definitely a history of enmity between them and Slavic Russians.
Going beyond that, this is a decent overview of the idea of Slavic identity and it's connection to Pan Slavism it cites a number of more details sources.
That source ties pan-slavism to the aftermath of the napoleonic war, and the desire of various slavic groups within eastern Europe to have independence. However, the movement was never really that cohesive because the poles didn't trust the Russians, the slovak groups were strongly aligned with the Austro-Hungarian empire against the Russians, and the Balkan slavs did align with the Russians against the Austro-Hungarian Empire (which, incidentally, led to the spark that ignited WWI).
Separately, it is important to note that Imperial Russia deliberately pushed the idea of pan-slavism to further Russian Imperial ends. That is, both liberating Slavic peoples to join the Russian empire, and encouraging the orthodox faith to spread in the same areas. You see this most strongly in Russian support of Serbian slavs, and Russian campaigns against the Ottomans. See also The all Russian empire The soviets co-opted this in some ways, in the way Russia came to be the primary player in the Soviet Union, although facially it was all covered up with socialist rhetoric.
That's not a very coherent answer, but just two thoughts on the idea of slavic identity.
I would question your premise.
Other linguistic/ethnic regions do show a lot of attempts at creating a common identity.
Two easy counter examples are pan-Arabism and La Raza.
Pan-Arabism, although it's easy to forget now, was the driving force of politics in the middle-east in the mid-twentieth century. Nasser was the most famous proponent of it, and pan-Arabism was one of the chief motivations for the ongoing Israeli/rest of the regions disputes. In 1958, the United Arab Republic briefly formed peacefully, showing more concrete progress towards pan-Arabism than a Slavic state has ever had. Post-1967, this has mostly been rolled back, but it's hard to make a case that pan-Slavism is really more of a thing than pan-Arabism was. Most of these nationalist movements have their rises and declines, and some are successful. Martin Kramer has a good discussion here: http://www.martinkramer.org/sandbox/reader/archives/arab-nationalism-mistaken-identity/
Also, you mention Germanic identity, but doesn't Bismark really put that statement in question? There isn't a push towards a pan-Germania because Bismark already accomplished it.
Related question: Does the strength of Slavic identity vary based on linguistic and religious factors. Some Slavic-speakers are Orthodox, some aren't. Some speak languages that are mutually intelligible with Russian, some don't. How much do those divisions matter?
An interesting factoid regarding this is that Proto-Slavic diverged into the modern Slavic languages much later than the proto-language of many other big language families, like Germanic or Romance. Where the Romance languages began to diverge substatially as early as 750, Proto-Slavic was spoken as a single linguistic unit, albeit with some variation from area to area as recently as 1150. This difference of 400 years, can make a difference in the depth of pan-linguistic identity.