Is the sea level rise unusual?
The problem is the increase in the rate of sea level rise. I pulled out some approximate numbers from the figure you presented:



Can you see now how the sea level is rising much faster today than a century ago?

Sea level rise, as well as climate change are normal things on Earth history. However, most times they happen at a very slow rate, allowing ecosystems and other processes to adapt to the change.

For example, if the climate changes over several hundreds of years, animals and even tree populations can "move" towards the side of their distribution where climate is still good for them. But if the climate changes in 50 years, all the trees can die before they had time to grow in the areas where they could thrive in the new climate.

In the case of sea level rise. The delta of a river for example, stays in equilibrium with the sea level because of the accumulation of sediments carried by the river. If the sea level rises slowly, the sediments can fill the delta and keep it roughly at sea level. But if the see level rises too quick for the sedimentation to keep up, it will be flooded by the sea, killing all the animals and people that live on such fertile environments.

Analogously, coastal infrastructure have a given lifetime. Let's say 50 years. If the sea level doesn't change much over that period (8.5 cm at the 1880-1940 rate), there is no problem. Once the infrastructure gets replaced, the new building will be set a bit higher.
However, in the next 50 years the sea level could rise 50 cm, or even more (it would be 19 cm if we assume the last rate from the figure won't increase any further), and that is a big deal. That could mean that much coastal infrastructure will be flooded, and maybe destroyed during storms.

In places like Bangladesh there are hundred of millions of people that could be displaced due to sea level rise if the rate keeps increasing. People that will also need to find a new home.

Coastal infrastructure loss, coastal erosion and immigration could be some of the worst expressions of fast sea level rise. 

ADDITION

Given the interest risen by this question I'm adding here some data beyond what is presented by the OP, and brought to my attention by @Bobson in the comments. It comes from the paper Recent global sea level acceleration started over 200 years ago? and shows a sea level reconstruction going back to 1700, and it shows also the changes in rate over that period. This is summarized in their figure 3:



Here you can clearly see how, with some ups and downs, the rate of sea level rise have been increasing over the last few centuries. And notably the current rate, about 20 years after the end of this plot is already out of the scale, and around 3.2 mm/year as pointed also by other answers.

I would highlight the following from their abstract:


  Sea level rose by 6 cm during the 19th century and 19 cm in the 20th
  century.Superimposed on the long-term acceleration are quasi-periodic
  fluctuations with a period of about 60 years. If the conditions that
  established the acceleration continue,then sea level will rise 34 cm
  over the 21st century. Longtime constants in oceanic heat content and
  increased ice sheet melting imply  that  the latest
  Intergovernmental Panel on Climate Change (IPCC) estimates of sea
  level are probably too low.


Regarding to whether this is caused by humans or not, I rather stay out of that argument and point that our best science and models suggest that lowering $\text{CO}_2$ emissions can make a significant impact in slowing down sea level rise in the upcoming centuries, so we should ACT NOW, and stop arguing whether it was or not our fault in the first place.