To complement the other answers, I would like to add a word on the analytic spectrum $\mathrm{Specan}(\mathbb{C}[[t]])$.

First, let me say that I am not sure what $\mathrm{Specan}$ means and have no idea where the notation comes from. On the other hand, it makes sense to consider the analytic spectrum of $\mathbb{C}[[t]]$ in R. Huber's theory of adic analytic spaces (where it would be called $\mathrm{Spa}(\mathbb{C}[[t]],\mathbb{C}[[t]])$) and in V. Berkovich's theory of analytic spaces (where it would be called $\mathcal{M}(\mathbb{C}[[t]])$ ; one would also need to prescribe the absolute of $t$, say $|t| = r \in (0,1)$, because the theory requires actual absolute values and not merely equivalence classes, *i.e.* valuations, but this is not really an issue).

Let me start with adic spaces. In this case, the spectrum is made of one closed point ($t=0$) and one open point (associated to the $t$-adic valuation). This open point is really the generic point of the space. More generally, you can associate (fully faithfully) an adic space to any sufficiently nice formal scheme and take its generic fiber inside the category of adic spaces. So here you really have the formal scheme again but, in the underlying space, you see its special fiber and its generic fiber (whereas $\mathrm{Spf}(\mathbb{C}[[t]])$ only shows the special fiber).

(Everything here looks quite similar to $\mathrm{Spec}(\mathbb{C}[[t]])$, so one may wonder why bother with formal schemes, fancy analytic spaces, etc. This is a only because the chosen situation is very simple (affine in particular) and you will have a hard time representing infinitesimal neighbourhoods by algebraic objects very quickly as soon as you start glueing.)

The situation with Berkovich spaces is very similar except that you will see a segment $[0,r]$ instead of two points. The point $0$ corresponds to $t=0$ and the other points all correspond to $t$-adic absolute values with different normalizations (given by $|t| = s$ for $s\in (0,r]$). Here again, you see a special fiber and a generic fiber (and even several equivalent copies of it). Note that the underlying space is Hausdorff and compact. This is a general feature of Berkovich spaces which is quite nice, although it is not clear how useful it would be in this particular situation.