As you can see here, there is plenty of what appears to be self-similarity within the graph of $ \tan{\left\lfloor{x}\right\rfloor}$. Every $7\pi$ interval along the x-axis there appears to be a repeating backwards tangent graph made from the integer flooring, but one can also see larger forwards tangent graphs.
What's the explanation behind all of this?
Actually, there is no self-similarity, the only reason why it looks self-similar is that $113\pi$ is about $354.9999698556$, extremely close to $355$ (In fact, both $\frac{22}{7}$ and $\frac{355}{113}$ are convergents of $\pi$). In fact, near zero, the maximums of each "period" grow at approximately $1.571$ per $355$ units. If you zoom out more, you'll see that those maximums create a forward tangent function with a period of $36997745$ (as $\frac{36997745}{1776748}$ differs from $\pi$ form less than 2 parts in a trillion), and the maximums of those tangents create another tangent, and so on. So, even if you were to use GeoGebra or mathway, you'll always see the "self-similarity".