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It is not genetic, and not hereditary. But, it is possible to breed lined brindled, it is just extremely unlikely. Brindles have a slightly bigger chance of producing brindle, but it is still a veeeery tiny chance.

Brindles in horses are typically a result of chimerism - twin foals combining in utero to create one horse. The different colors would have different DNA, and that would extend to any organs contributed by one foal versus the other.

Conceiving twins has a slight genetic component, so brindle is heritable in that regard - a mare whose dam conceived twins (even if they later merged to create a brindle) is more likely to conceive twins herself. (This happens in cows and humans, too.) Since conception of twins is a precondition for brindle, being more likely to conceive twins makes you more likely to produce brindle. But twins, and the chimerism that results in brindle are very low probability in horses. @Maribo really sums it up the best. "Brindles have a slightly bigger chance of producing brindle, but it is still a veeeery tiny chance."