Let $$dY_t(\omega )=\mu(t,\omega )dt+\sigma (t,\omega )dB_t(\omega ).\tag{1}$$
Why exactly $(dY_t)^2$ is the quadratic variation of $dY_t$ ? At the end, what means exactly $dY_t$ for a process $Y_t$ ? Indeed, here $(1)$ is a notation for $$Y_t=Y_0+\int_0^t\mu(s,\cdot )ds+\int_0^t \sigma (s,\cdot )dB_s,$$
1) but avoiding $(1)$, what would be $dY_t$ in general ?
2) And also, what is $(dY_t)^2$ ? Why should it be a quadratic variation of the obscure process $dY_t$ ?