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In contrast, for plurisubharmonic functions and their subextensions the situation is more difficult: In: Fornæss, John Erik; Sibony, Nessim: Plurisubharmonic functions on ring domains. Complex analysis (University Park, Pa., 1986), 111–120, Lecture Notes in Math., 1268, Springer, Berlin, 1987 (MR0907057), the authors consider plurisubharmonic (psh) function on $B\setminus K$, where $B$ is a ball in $\mathbb{C}^n$ and $K$ is a polynomially convex set. They give the following counterexamples: one is a smooth psh function on $B\setminus K$, $K$ a polydisk, which does not have a psh subextension to $B$; another one is a discontinuous psh function on $B\setminus K$, $K$ a smaller ball, which does not have a psh subextension to $B$. There is also earlier work about psh functions that cannot be extended across a pseudoconvex set: Bedford, Eric; Burns, Dan Domains of existence for plurisubharmonic functions. Math. Ann. 238 (1978), no. 1, 67–69 (MR0510308).

Finally, as I mentioned earlier, even if extensions are possible, not all of them are pluriharmonic: take $M=\mathbb{C}^2$, $\Omega=\{(z,w):|z|^2+|w|^2\leq R\}$, with large $R$, and $u(z,w)=\max\{\log^+|z|,\log^+|w|\}$. Then $u$ is pluriharmonic on $\mathbb{C}^2\setminus \Omega$, but it is not a pluriharmonic function on $\mathbb{C}^2$ (since $dd^cu\wedge dd^cu$ is a probability measure supported on $\{|z|=|w|=1\}$).

In contrast, for plurisubharmonic functions and their subextensions the situation is more difficult: In: Fornæss, John Erik; Sibony, Nessim: Plurisubharmonic functions on ring domains. Complex analysis (University Park, Pa., 1986), 111–120, Lecture Notes in Math., 1268, Springer, Berlin, 1987 (MR0907057), the authors consider plurisubharmonic (psh) function on $B\setminus K$, where $B$ is a ball in $\mathbb{C}^n$ and $K$ is a polynomially convex set. They give the following counterexamples: one is a smooth psh function on $B\setminus K$, $K$ a polydisk, which does not have a psh subextension to $B$; another one is a discontinuous psh function on $B\setminus K$, $K$ a smaller ball, which does not have a psh subextension to $B$. There is also earlier work about psh functions that cannot be extended across a pseudoconvex set: Bedford, Eric; Burns, Dan Domains of existence for plurisubharmonic functions. Math. Ann. 238 (1978), no. 1, 67–69 (MR0510308).

Even for plurisubharmonic functions and their subextensions the situation is difficult: In: Fornæss, John Erik; Sibony, Nessim: Plurisubharmonic functions on ring domains. Complex analysis (University Park, Pa., 1986), 111–120, Lecture Notes in Math., 1268, Springer, Berlin, 1987 (MR0907057), the authors consider plurisubharmonic (psh) function on $B\setminus K$, where $B$ is a ball in $\mathbb{C}^n$ and $K$ is a polynomially convex set. They give the following counterexamples: one is a smooth psh function on $B\setminus K$, $K$ a polydisk, which does not have a psh subextension to $B$; another one is a discontinuous psh function on $B\setminus K$, $K$ a smaller ball, which does not have a psh subextension to $B$. There is also earlier work about psh functions that cannot be extended across a pseudoconvex set: Bedford, Eric; Burns, Dan Domains of existence for plurisubharmonic functions. Math. Ann. 238 (1978), no. 1, 67–69 (MR0510308).

Finally, as I mentioned earlier, even if extensions are possible, not all of them are pluriharmonic: take $M=\mathbb{C}^2$, $\Omega=\{(z,w):|z|^2+|w|^2\leq R\}$, with large $R$, and $u(z,w)=\max\{\log^+|z|,\log^+|w|\}$. Then $u$ is pluriharmonic on $\mathbb{C}^2\setminus \Omega$, but it is not a pluriharmonic function on $\mathbb{C}^2$ (since $dd^cu\wedge dd^cu$ is a probability measure supported on $\{|z|=|w|=1\}$).

In contrast, for plurisubharmonic functions and their subextensions the situation is more difficult: In: Fornæss, John Erik; Sibony, Nessim: Plurisubharmonic functions on ring domains. Complex analysis (University Park, Pa., 1986), 111–120, Lecture Notes in Math., 1268, Springer, Berlin, 1987 (MR0907057), the authors consider plurisubharmonic (psh) function on $B\setminus K$, where $B$ is a ball in $\mathbb{C}^n$ and $K$ is a polynomially convex set. They give the following counterexamples: one is a smooth psh function on $B\setminus K$, $K$ a polydisk, which does not have a psh subextension to $B$; another one is a discontinuous psh function on $B\setminus K$, $K$ a smaller ball, which does not have a psh subextension to $B$. There is also earlier work about psh functions that cannot be extended across a pseudoconvex set: Bedford, Eric; Burns, Dan Domains of existence for plurisubharmonic functions. Math. Ann. 238 (1978), no. 1, 67–69 (MR0510308).

Finally, as I mentioned earlier, even if extensions are possible, not all of them are pluriharmonic: take $M=\mathbb{C}^2$, $\Omega=\{(z,w):|z|^2+|w|^2\leq R\}$, with large $R$, and $u(z,w)=\max\{\log^+|z|,\log^+|w|\}$. Then $u$ is pluriharmonic on $\mathbb{C}^2\setminus \Omega$, but it is not a pluriharmonic function on $\mathbb{C}^2$ (since $dd^cu\wedge dd^cu$ is a probability measure supported on $\{|z|=|w|=1\}$).

Here is a counterexample with bounded $\Omega$: take $m=2$, $\Omega=\{(z,w):|z|^2+|w|^2\leq R\}$, where $R>1$, and $u(z,w)=\max\{\log^+|z|,\log^+|w|\}$. Then $u$ is pluriharmonic on $\mathbb{C}^2\setminus \Omega$, but it is not a pluriharmonic function on $\mathbb{C}^2$ ($dd^cu\wedge dd^cu$ is a probability measure supported on $\{|z|=|w|=1\}$).

Edit: this is not quite a counterexample you are looking for; I was trying to point out that even if extensions are possible, not all of them are pluriharmonic in the whole space. I am making my answer a CW until a more precise one is found.

Even for plurisubharmonic functions and their subextensions the situation is difficult: In: Fornæss, John Erik; Sibony, Nessim: Plurisubharmonic functions on ring domains. Complex analysis (University Park, Pa., 1986), 111–120, Lecture Notes in Math., 1268, Springer, Berlin, 1987 (MR0907057), the authors consider plurisubharmonic (psh) function on $B\setminus K$, where $B$ is a ball in $\mathbb{C}^n$ and $K$ is a polynomially convex set. They give the following counterexamples: one is a smooth psh function on $B\setminus K$, $K$ a polydisk, which does not have a psh subextension to $B$; another one is a discontinuous psh function on $B\setminus K$, $K$ a smaller ball, which does not have a psh subextension to $B$. There is also earlier work about psh functions that cannot be extended across a pseudoconvex set: Bedford, Eric; Burns, Dan Domains of existence for plurisubharmonic functions. Math. Ann. 238 (1978), no. 1, 67–69 (MR0510308).

Finally, as I mentioned earlier, even if extensions are possible, not all of them are pluriharmonic: take $M=\mathbb{C}^2$, $\Omega=\{(z,w):|z|^2+|w|^2\leq R\}$, with large $R$, and $u(z,w)=\max\{\log^+|z|,\log^+|w|\}$. Then $u$ is pluriharmonic on $\mathbb{C}^2\setminus \Omega$, but it is not a pluriharmonic function on $\mathbb{C}^2$ (since $dd^cu\wedge dd^cu$ is a probability measure supported on $\{|z|=|w|=1\}$).