Slicer isn't filtering data properly (with sample)

It does filter correctly but in your mesure you are not filtering the fact_messages based on the date, you are filtering the messages based on the relationships employee_id.

You probably want to add a filter on the date_created column of the fact_messages:

MessageCount = 
VAR SelectedUnitName = SELECTEDVALUE(dim_org[unit_name])
RETURN
CALCULATE(
    COUNTROWS(fact_messages),
    FILTER(
        dim_employees,
        dim_employees[start_date] <= MAX(dim_date[Date])
            && dim_employees[end_date] >= MIN(dim_date[Date])
    ),
    FILTER(
        dim_org,
        dim_org[unit_name] = SelectedUnitName
    ),
    FILTER(
        fact_messages,
        fact_messages[date_created] <= SELECTEDVALUE(dim_employees[end_date])
           && fact_messages[date_created] >= SELECTEDVALUE(dim_employees[start_date])
    ),
)

 Not sure if I have that syntax correct in the filter I have added but I think you get the idea ?