Let $f: N \to M$ be a smooth immersion and let $p \in M$, $W = f(V) \subset M$ be an submanifold with $q = f(p).$
Then the sequence is split exact
$$T_qW \hookrightarrow T_qM \stackrel{\mu}\to T_qM/T_qW$$
Therefore $T_qM \cong T_qW \oplus T_qM/T_qW.$ Now apparently the quotient according to wikipedia on normal bundles, it can be identified with the normal space (as a fiber) $N_qW.$ That is $NM \stackrel{\pi}\to M$, $$N_qW = T_qM/T_qW = \pi^{-1}(p).$$
I don't know how this last conclusion is reached...
The only answer I am thinking of is that there is some surjective linear (canonical?) map $\ell: T_qW \to N_qW$, then it would yield the commutative diagram:
$$\begin{array}{cccccccc} T_qM & \xrightarrow{\mu} & T_qM/T_qW & \\ \downarrow & \swarrow \\ N_qW \end{array}$$
Actually I just realized what is really going on, the answer is almost silly now that I realize it.
Since $T_qW \subset T_qM$, we can talk about its orthogonal complement $N_q(W)$ which has direct sum with $T_qW \oplus N_qW \approx T_qW \oplus T_qM/T_qV.$ By comparing the elements, it is not surprising that there is an identification between the spaces in question. I just don't know what it is, but my initial question appears to be answered.