This question is directly related to when $d(n)=d(n+1)$ where $d(n)$ denotes the divisor function.

**Solutions to $d(n)=d(n+1)$:**

In 1952, Erdos and Mirsky conjectured that $d(n)=d(n+1)$ has infinitely many solutions.  In 1984, Heath Brown proved this result, and gave a lower bound on the counting function.  Let   Let $\widetilde{D}(x)$ denote the number of $n\leq x$ satisfying  $d(n)=d(n+1)$.  Heath Brown showed that  $$\widetilde{D}(x)$\gg \frac{x}{(\log x)^7}.$$

In 1987 Erdős, Pomerance and Sárközy gave the upper bound $$\widetilde{D}(x)\ll \frac{x}{(\log \log x)^\frac{1}{2}}.$$

Later that year, Hildebrand improved Heath Browns Result that $$\widetilde{D}(x)\gg \frac{x}{(\log \log x)^3},$$ showing that the correct magnitude involves a doubly logarithmic factor.

**Consecutive integers with identical prime signature:**

Note that Erdős, Pomerance and Sárközy result immediately implies that your counting function is $$\ll \frac{x}{(\log \log x)^\frac{1}{2}},$$ and so it is not linear even though the graph resembles a line.  This is because $\log \log x$ grows extremely slowly.

It seems likely that one could use Hildebrands lower bound to prove that the set of consecutive integers with identical prime signature is infinite.  This is because $d(n)=d(n+1)$ "often" implies that $n$ and $n+1$ have the same signature. 

**Some References:** (Chronological Ordering)

 - Erdös, Mirsky 1952: [The distribution of the values of $d(n)$.][1]

 - Heath-Brown 1984: [The divisor function at consecutive integers.][2]

 - Erdős, Pomerance and Sárközy 1987 [On locally repeated values of certain arithmetic functions. III.][3] 

 - Hildebrand 1987: [The divisor function at consecutive integers.][4]


  [1]: http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=49932&loc=fromrevtext
  [2]: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6990892
  [3]: http://www.jstor.org/stable/2046541?origin=crossref
  [4]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102690578