You could try changing from an EventTrigger to a MultiTrigger and using the IsMouseOver property rather than the MouseEnter event. However, as you discovered, Storyboards in styles do not allow you to specify a TargetName, so it's important to move the triggers to the target object.
The example below hard-codes the value of Rotor to 1 and you'll need to correctly namespace the RenderTransform in the TargetProperty or you'll get a runtime exception.
<Grid>
<Grid.Resources>
<Storyboard x:Key="ShakeStatorMinorRadiusEdit">
<DoubleAnimationUsingKeyFrames Storyboard.TargetProperty="RenderTransform.(TranslateTransform.X)"
RepeatBehavior="5x">
<EasingDoubleKeyFrame KeyTime="0:0:0.05"
Value="0" />
<EasingDoubleKeyFrame KeyTime="0:0:0.1"
Value="3" />
<EasingDoubleKeyFrame KeyTime="0:0:0.15"
Value="0" />
<EasingDoubleKeyFrame KeyTime="0:0:0.20"
Value="-3" />
<EasingDoubleKeyFrame KeyTime="0:0:0.25"
Value="0" />
</DoubleAnimationUsingKeyFrames>
</Storyboard>
</Grid.Resources>
<Grid.ColumnDefinitions>
<ColumnDefinition />
<ColumnDefinition />
<ColumnDefinition />
<ColumnDefinition />
</Grid.ColumnDefinitions>
<Grid.RowDefinitions>
<RowDefinition Height="Auto" />
<RowDefinition Height="Auto" />
<RowDefinition Height="Auto" />
<RowDefinition Height="Auto" />
</Grid.RowDefinitions>
<TextBlock Grid.Column="1">Rotor</TextBlock>
<TextBlock Grid.Column="2">Stator</TextBlock>
<TextBlock Grid.Column="0"
Grid.Row="1">Lobes</TextBlock>
<TextBlock Grid.Column="0"
Grid.Row="2">Major Radius</TextBlock>
<TextBlock Grid.Column="0"
Grid.Row="3">Minor Radius</TextBlock>
<TextBox Name="RotorLobes"
Grid.Column="1"
Grid.Row="1"
Text="1" />
<TextBox Grid.Column="1"
Grid.Row="2" />
<TextBox Name="MinorRadiusRotor"
Background="Blue"
Grid.Column="1"
Grid.Row="3" />
<TextBox Grid.Column="2"
Grid.Row="1" />
<TextBox Grid.Column="2"
Grid.Row="2" />
<TextBox Name="MinorRadiusStator"
Background="Green"
Grid.Column="2"
Grid.Row="3">
<TextBox.Style>
<Style>
<Style.Triggers>
<MultiDataTrigger>
<MultiDataTrigger.Conditions>
<Condition Binding="{Binding ElementName=MinorRadiusRotor, Path=IsMouseOver}"
Value="True" />
<Condition Binding="{Binding ElementName=RotorLobes, Path=Text}}"
Value="1" />
</MultiDataTrigger.Conditions>
<MultiDataTrigger.EnterActions>
<BeginStoryboard Storyboard="{StaticResource ShakeStatorMinorRadiusEdit}" />
</MultiDataTrigger.EnterActions>
</MultiDataTrigger>
</Style.Triggers>
</Style>
</TextBox.Style>
</TextBox>
</Grid>