The answer should be set PerformanceGoal -> "Quality". Otherwise PerformanceGoal options is set to global $PerformanceGoal which switches to "Speed" if the evaluation is done on a preemptive link. (default for most of gui/Dynamic etc).
See more in linked duplicate.
However, there appears to be a bug in 11+ which makes this option being ignored, fortunately Mathematica is great in workingaround bugs it introduces. So instead using the option we can Block the $PerformanceGoal:
Animate[
Block[{$PerformanceGoal = "Quality"}
, Plot3D[Sin[t x y], {x, -3, 3}, {y, -3, 3}, PlotRange -> {0, 1.2}]
]
, {t, 0, 2}
, AnimationRunning -> False
, DisplayAllSteps -> True
]
