Yes, for odd number of variable XOR gate behaves similar to XNOR gate.
- Output of $\oplus$ is $1$ for odd number of $1's$
Ex: $0\oplus1\oplus1\oplus1\oplus0 = 1$ (odd number of 1's)
- Output of $\odot$ is $1$ for even number of $0's$
Ex: $1\odot1 = 1$ (0 number of $0's$) and $1\odot0\odot0\odot1 = 1$(even $0's$)
Input to a gate can either be $0$ or $1$. So, If the number of input variables is odd. then either $0's$ as input are odd and $1's$ as input are even (or) $0's$ are even and $1's$ are odd.
I mean to say if a XOR gate takes odd number of inputs, then either 0's are odd (or) $1's$ are odd.
If number of input variables is odd and $1's$ in it are odd, then output of XOR gate is $1$ (remember XOR gives $1$ for odd number of $1's$ in input), and if $1's$ are odd, then it contains even number of $0's$, which implies that output of XNOR is also $1$(XNOR gives $1$ for even number of zeros)
If number of input variables is odd and $0's$ are odd, then output of XNOR is $0$ (XNOR is $1$ for even number of $0's$), thus number of $1's$ in input is even, and for even number of $1's$ in input XOR gives $0$.
Although it has become bit complicated but It can be seen from above two cases that when number of input variables to a XOR and XNOR gate are odd, they behave exactly same.
So, for even number of inputs $\color{red}{\oplus = \overline{\odot}}$ and odd number of inputs $\color{red}{\oplus = \odot}$
