Why do we pretend + and x not to be addition and multiplication while eventually they are considered so while explaining groups, rings and fields? Aren't commutative, associative properties result of addition and multiplication? Then why most of the definitions of Groups and Fields contain + and x but don't accept that they are addition and multiplication?

As an exemple of how '' addition'' and ''multiplication'' can be different from the usual operations in $\mathbb{R}$, take the ring $(M_2(\mathbb{R},\oplus, \odot)$ of $2\times 2$ matrices with real entries.
Here the $\oplus$ is defined as
$$
A\oplus B=
\left(
 \begin{array}{ccccc}
a_1&a_2 \\
a_3&a_4  
\end{array}
 \right) \oplus
\left(
 \begin{array}{ccccc}
b_1&b_2 \\
b_3&b_4  
\end{array}
 \right) =
 \left(
 \begin{array}{ccccc}
a_1+b_1&a_2+b_2 \\
a_3+b_3&a_4+b_4  
\end{array}
 \right) 
$$
and we can easly see that this operation is associative, commutative, has a neutral element (the matrix with all $0$ entries, that we call $0$) and that every matrix $A$ has an ''inverse $X$'' (that we call ''opposite'') such that $A \oplus X=0$ ( and we call this opposite $-A$), i.e. $(M_2(\mathbb{R}),\oplus)$ is a commutative group.
Is all that sufficient to call $\oplus$ a ''sum'' in analogy whit the sum in $\mathbb{R}$? NO, because also the product in $\mathbb{R}$ has the same formal properties.
But we also define the operation $\odot$ as:
$$
A\odot B=
\left(
 \begin{array}{ccccc}
a_1&a_2 \\
a_3&a_4  
\end{array}
 \right) \odot
\left(
 \begin{array}{ccccc}
b_1&b_2 \\
b_3&b_4  
\end{array}
 \right) =
 \left(
 \begin{array}{ccccc}
a_1b_1+a_2b_3&a_1b_2+a_2b_4 \\
a_3b_1+a_4b_3&a_3b_2+a_4b_4  
\end{array}
 \right) 
$$
and we see that such opertion is associative, but not commutative, there is a neutral element, the matrix
$$
I=
\left(
 \begin{array}{ccccc}
1&0 \\
0&1  
\end{array}
 \right)
$$
but there are matrices $M \ne 0$ that don't have an ''inverse'', as:
$$
M=
\left(
 \begin{array}{ccccc}
0&m \\
0&0  
\end{array}
 \right)
$$
So $(M_2(\mathbb{R}),\odot)$ is not a group but only a monoid.
So the first operation $\oplus$ has the same formal properties of $+$ and $\cdot$ in $\mathbb{R}$, but the second operation $\odot$ has not these properties. Why ve call the first a ''sum'' and the second a ''product''? Because there is another important property: $\odot$ is distributive over $\oplus$, i.e.:
$$
A\odot(B\oplus C)=a\odot B \oplus A\odot C
$$
And here $\oplus$ and $\odot$ have the same behavior as $+$ and $\cdot$.
So $\oplus$ is very similar to the $+$ operation on $\mathbb{R}$ and we can call it a ''sum''. Finally we chose to  call $\odot$ a ''product'' ( or ''multiplication'')  only to save words, but we we must be careful that this ''product'' is not commutative and not invertible.
