I have done this experiment plenty of times with my students. It is one of my favorites. The results are surprising, and often difficult to reproduce. It depends on the current. With a low current $(I < \pu{0.05 A}),$ long, bright and thin metallic needles are obtained starting from the cathode. They sometimes touch one another in the middle of the box, like in your experiment.
Anyway, after a couple of minutes, the needles grow so much that they join the electrodes. This must be avoided, because it stops the electrolysis process. In this case, the “bridge” should be broken by hand with a glass stick.
If the current increases $(I > \pu{0.1 A}),$ the needles are not so thin. On the contrary, they tend to be tightly packed. If the current is rather high $(I > \pu{0.2 A}),$ they even look like gray and dull moss… Deceiving!
On the anode, the $\ce{Sn^{2+}}$ ion is oxidized into $\ce{Sn^{4+}}$ according to
$$\ce{Sn^{2+} -> Sn^{4+} + 2 e-},\tag{R1}$$
but this ion is quickly hydrolyzed into weekly acidic solution, producing a white deposit of $\ce{SnO2·nH2O}$ according to
$$\ce{Sn^{4+} + $(2 + n)$ H2O -> SnO2·nH2O + 4 H+}.\tag{R2}$$
It is possible to prevent this white precipitate by dissolving $\ce{SnCl2}$ in a $\pu{2 mol/L}$ $\ce{HCl}$ solution, which reverses the last equation.
Something interesting can be observed at the end, when you think the experiment is finished. At this moment, be careful. Don't touch the solution, but disconnect the wires connecting the electrodes to the power supply. And quickly insert these plugs into a $\pu{50 mA}$ ammeter. You'll see that some current is produced for a couple of seconds. Your small tray is working like a galvanic cell!
Last but not least. At the end, when the tin has invaded the whole solution (except a tiny zone around the anode), try to save the tin from the tray with your hand (gloves!). Press it on a blotting paper or a filter paper. When dry put it in a tiny test tube, and heat it with a Bunsen burner. It quickly melts, and the liquid can be thrown on the lab bench: it makes a bright and shiny metallic stain or platelet. Looks like silver! Now try to bend this thin platelet with your hands near the ear. It makes a surprising noise, showing that bending the tin platelet breaks tiny tin crystals.
(Left hand side)
(In the middle, it has dendritic metal on its right, and little amount of white precipitate on its left)
