The simple solution as already given in the comments is to split the Line
expression and interleave color directives:
Animate[Graphics[{{Red, Line[{{0, 0}, {3 Cos[x], 3 Sin[x]}}]},
{Blue, Line[{{0, 0}, {3 Cos[x - x/10], 3 Sin[x - x/10]}}]}}, Axes -> True,
PlotRange -> {{-5, 5}, {-5, 5}}], {x, 0, 2 π}]
In the case of more line segments there can be an advantage to preserving a single Line
expression as this is typically rendered faster. To make that work with individual segment colors you can use VertexColors
and specify the color for each end of each line segment:
Animate[Graphics[
Line[{{{0, 0}, {3 Cos[x], 3 Sin[x]}}, {{0, 0}, {3 Cos[x - x/10],
3 Sin[x - x/10]}}}, VertexColors -> {{Red, Red}, {Blue, Blue}}], Axes -> True,
PlotRange -> {{-5, 5}, {-5, 5}}], {x, 0, 2 π}]

A Larger example:
colors = ({#, #} & /@ RandomColor[50]);
Animate[
Graphics[
Line[{#, {0, 0}} & /@ CirclePoints[{1, x}, 50], VertexColors -> colors]
],
{x, 0, Pi/5}
]

Graphics
command likeGraphics[{color1, line1, color2, line2}]
$\endgroup$Animate[ Graphics[ Thread[{{Red, Blue}, Line /@ {{{0, 0}, {3 Cos[x], 3 Sin[x]}}, {{0, 0}, {3 Cos[x - x/10], 3 Sin[x - x/10]}}}}], Axes -> True, PlotRange -> {{-5, 5}, {-5, 5}}], {x, 0, 2 \[Pi]}]
$\endgroup$