There is a standard functionality on the Sheet side like the Dashboard actions called "Highlighter".
Take your Line Card > Right Click > Highlight Selected Items.
You can also modify and create an other Worksheet Action with the menu > Worksheet > Actions
What version of Tableau are you using? Is your data set static, or will the data get refreshed fairly often (daily, weekly, etc)?
I'm halfway to a solution, but it's very manual and fairly tortured. See here: https://public.tableau.com/profile/jennifer.vonhagel#!/vizhome/HoverReferenceLine/Dashboard1?publish=yes
I created a second worksheet called HoverBars that contain thin bars at each date as well as the labels for the trend line points. Since Tableau 2018.3 has transparent worksheets, I put your main data chart on top of the HoverBars worksheet in a dashboard, aligned them perfectly, set a Hover action filter so that when a point is hovered on in your main data chart, it filters for the date only on the Hover chart, showing that date's bar and data label.
1. You have to Tableau 2018.3 to make use of transparent worksheets
2. You have to manually "fix" both the value and date axes of both charts. So when the data is refreshed, you may need to manually update the fixed axes. The axes have to be fixed so that the dates and values align when one chart is sitting on top of the other chart.
3. The big colored labels that appear when hovering over a date are BEHIND the main chart. This means all the dots will be over the label. If the label is very bold and the dots are mostly transparent, this may work, but it's pretty unsatisfying as a solution.
4. The charts have to be floating on the dashboard, and you have to place them very precisely so their dates and values align. Your dashboard can't be allowed to Auto-size, it will need to remain a Fixed Size so the alignments of the floating charts stay in sync.
Dashboard with Hover Line and Labels
I've attached the workbook. If you have questions or would like me to detail the steps better here I'd be happy to, but it might make sense to first learn whether you can (or are willing to ) accommodate all the necessary conditions. If nothing else, maybe this will spawn ideas for a more streamlined solution.
Hover Reference Line.twbx 29.7 KB
I mocked this up with parameters and a calculated field - is this what you are looking for ?
n.b. you use the date parameter to change the date value that is displayed...
Book2 (2).twbx 19.1 KB
Create chart with contentious axis date and measure.
Add ref line in date axis.
Create highlight action to add source and target with selected date field with hover option.
Find attached workbook.
Moving ref line.twbx 1.5 MB