Is '$10$' a magical number or I am missing something?
Short answer: your confusion about whether ten is special may come from reading aloud "Every base is base 10" as "Every base is base ten" — this is wrong; not every base is base ten, only base ten is base ten. It is a joke that works better in writing. If you want to read it aloud, you should read it as "Every base is base one-zero".
You must distinguish between numbers and representations. A pile of rocks has some number of rocks; this number does not depend on what base you use. A representation is a string of symbols, like "10", and depends on the base. There are "four" rocks in the cartoon, whatever the base may be. (Well, the word "four" may vary with language, but the number is the same.) But the representation of this number "four" may be "4" or "10" or "11" or "100" depending on what base is used.
The number "ten" — the number of dots in ".........." — is not mathematically special. In different bases it has different representations: in base ten it is "10", in base six it is "14", etc.
The representation "10" (one-zero) is special: whatever your base is, this representation denotes that number. For base $b$, the representation "10" means $1\times b + 0 = b$.
When we consider the base ten that we normally use, then "ten" is by definition the base for this particular representation, so it is in that sense "special" for this representation. But this is only an artefact of the base ten representation. If we were using the base six representation, then the representation "10" would correspond to the number six, so six would be special in that sense, for that representation.
The magic of the number 10 comes from the fact that "1" is the multiplicative unit and "0" is the additive unit. The first two-digit-number in positional notation is always 10 and also always denotes the number of digits.
Yes, ten ( ..... ..... ) is a special number. Not magical but special because it is a very convenient base for species that have ten fingers.
Arguably we can use hands and fingers to encode 1024 numbers using the binary system, but that would be less robust across reading directions and some configurations/gestures are physiologically hard to do.