Are humans the only species who drink milk as adults?
Good observation!
Gene coding for the lactase
Gene LCT
Mammals have a gene (called LCT C/T-13910) coding for the lactase enzyme, a protein able to digest lactose. Lactose is a disaccharide sugar found in milk.
Expression of LCT
In mammals, the gene LCT is normally expressed (see gene expression) only early in development, when the baby feeds on his/her mother's milk. Some human lineages have evolved the ability to express LCT all life long, allowing them to drink milk and digest lactose at any age.
Today, the inability to digest lactose at all ages in humans is called lactose intolerance.
Evolution of lactose tolerance in human
Three independent mutations
Tishkoff et al. 2007 found that the ability to express LCT at an old age has evolved at least three times independently. Indeed, they found three different SNPs (stands for Single Nucleotide Polymorphism; it is a common type of mutation), two of them having high prevalence in Africa (and people of African descent) and one having high prevalence in Europe (and people of European descent). The three SNPs are G/C-14010, T/G-13915 and C/G-13907.
Pastoralist populations
Lactose tolerance is much more common in people descending from pastoralist populations than in people descending from non-pastoralist populations, suggesting a strong selection for lactose tolerance Durham 1991.
Selective sweep
On top of that, Tishkoff et al. 2007 focusing on the locus 14010 (one of the three SNP's mentioned above) showed that there is a clear selective sweep (which is a signature of past and present selection) around this locus.
They estimated the age of the allele allowing lactose tolerance at this locus (allele C is derived, the ancestral being G; see nucleotide) at around 3,000 to 7,000 years (with a 95% confidence interval ranging from 1,200 to 23,200 years) and a selection coefficient of 0.04 - 0.097 (with a 95% confidence interval ranging from 0.01 to 0.15).
I recommend reading Tishkoff et al. 2007. It is a classic, is short and is relatively easy to read, even for someone with only basic knowledge in evolutionary biology.
Are humans the only animal that is able to drink milk as adults?
I don't really know... but I would think so, yes!
Drink vs digest thoroughly
As @anongoodnurse rightly said in his/her answer

&quot;Drink&quot; and &quot;digest thoroughly&quot; are two different things

Pets
According to many dog health websites (such this one for example) claim that there is also variance among dogs where some dogs are lactose tolerant and others are lactose intolerant. I could not find any paper on the underlying genetics of lactose intolerance in dogs or other pets. It is not impossible our pets have also been under selection to be able to digest lactose as we humans could have given milk to them. It is also possible that pets do not actually produce any lactase at adult age but rather that some pets are just able to deal with having indigestible lactose in their guts! But then again, &quot;Drink&quot; and &quot;digest thoroughly&quot; are two different things.
Tits and robins in 20th century England
A funny and famous case is the case of blue tits and robins in the 20th century, in England. At that time, in England, the milkman was bringing the milk at home in the morning and would leave glass bottles with a simple aluminum cap in front of people's home. At some point, blue tits and robins learnt that by pecking through the aluminum they can get access to the milk. See this (non-peer-reviewed) article that tells the story.
Somewhat related
There are already a number of good posts on milk digestion in humans on Biology.SE. Consider having a look at:

What inactivates pepsin in infants? and Seriously, do humans produce rennin?
Are there any non-mammalian species known that lactate?
Can an adult without genetic lactase persistence still develop a tolerance for dairy foods?