You're writing a book or going through an existing book?

Everything you wrote looks good except for one thing. For discrete distributions there is a dx value that is needed. However, that doesn't seem fix things.

The following seems to get a bit farther:

f[{x1_, x2_}] := Piecewise[{{1/3, {x1, x2} == {1, 1}}, {2/3, {x1, x2} == {2, 0}}}];
\[ScriptCapitalD] = ProbabilityDistribution[f[{x1, x2}], {x1, -10, 10, 1}, {x2, -10, 10, 1}];
Mean[\[ScriptCapitalD]]

$$\left\{\frac{5}{3},\frac{1}{3}\right\}$$

Variance[\[ScriptCapitalD]]

$$\left\{2/9, 2/9\right\}$$

Covariance[\[ScriptCapitalD]]

$$\left\{\left\{2/9, -(2/9)\right\}, \left\{-(2/9), 2/9\right\}\right\}$$

\[ScriptCapitalD]2 = TransformedDistribution[2 {x1, x2}, {x1, x2} \[Distributed] \[ScriptCapitalD]] // 
   FullSimplify;
Mean[\[ScriptCapitalD]2]

$$\left\{\frac{10}{3},\frac{2}{3}\right\}$$

Variance[\[ScriptCapitalD]2]

$$\left\{\frac{8}{9},\frac{8}{9}\right\}$$

Covariance[\[ScriptCapitalD]2]

$$\left\{\left\{8/9, -(8/9)\right\}, \left\{-(8/9), 8/9\right\}\right\}$$

But RandomVariate doesn't work with the above. For your particular example things can be rewritten so that RandomVariate works

d = TransformedDistribution[2 {x + 1, 1 - x}, x \[Distributed] BernoulliDistribution[2/3]];
RandomVariate[d, 10]
(* {{4, 0}, {2, 2}, {4, 0}, {4, 0}, {4, 0}, {4, 0}, {4, 0}, {2, 2}, {4, 0}, {4, 0}} *)

I suspect that ProbabilityDistribution and/or TransformedDistribution just can't handle the structure you have. (I'd like to be wrong about that.)

The dx term is found in the online reference for ProbabilityDistribution.

From the online reference:

No. Read the online reference of Discrete Multivariate Distributions under Scope. You still need dx.