You cannot specify a Date field for "Fetch Criteria" setting in the "Related records".
To retrieve records that contain the specified date value and display them in the Related records field, you need to use Calculated or Text field for "Fetch Criteria" setting instead.
Reference: Fields that Can Be Specified for the "Fetch Criteria" Setting in Related Records

(1) In a Calculated or Text field, set a formula that refers to a value in a date field
(2) In "Related records" field, specify the Calculated or Text field you configured for the "Fetch Criteria"

In the following example, we explain how to configure "Lead Management" app from which retrieve data, and "Customer Management" app to place a Related records field.

1. Add a Date field in the Lead Management app.
   Leave the Default Value as it is.

2. In the Lead Management app, add a Calculated or Text field to get the Date field value.

   • When using a "Calculated" field:

     • Field Name: Date (Calculated)
     • Field Code: Date_Calculated
     • Formula: Date
     • Display format: Date (Example: Aug 06, 2012)
       

   • When using a "Text" field:

     • Field Name: Date (Text)
     • Field Code: Date_Text
     • Formula: DATE_FORMAT(Date,"MM-dd-YYYY","Etc/GMT")
       

3. If there are existing records in the app, update them as needed.
   You can also bulk update records by performing the Export to File and Import from File actions.
   Exporting Data to a File
   Importing Record Data from a File into an App

1. Place a Date field in the Customer Management app as well.
   Leave the Default Value as it is.

2. In the Customer Management app, add a Calculated or Text field to get the Date field value.

   • When using a "Calculated" field:

     • Field Name: Date (Calculated)
     • Field Code: Date_Calculated
     • Formula: Date
     • Display format: Date (Example: Aug 06, 2012)
       

   • When using a "Text" field:

     • Field Name: Date (Text)
     • Field Code: Date_Text
     • Formula: DATE_FORMAT(Date,"MM-dd-YYYY","Etc/GMT")
       

3. If there are existing records in the Customer Management app, update them as needed.
   You can also bulk update records by performing the Export to File and Import from File actions.
   Exporting Data to a File
   Importing Record Data from a File into an App

4. Add a Related records field.
   Specify the Calculated or Text field that you configured in Step 2 for the app from which you retrieve data.

   • Field Name: Related records
   • Field Code: Related_Records
   • Datasource App: Lead Management
   • Fetch Criteria: "Date (Calculated)" = "Date (Calculated)" or "Date (Text)" = "Date (Text)"
   • Datasource App Fields to Display: Optional
     Refer to Configuring a "Related Records" Field

You have now completed the settings.

1. In the Lead Management, add a new record with a date.
   The same value is automatically displayed on the "Date (Calculated)" or "Date (Text)" field.

2. In the Customer Management app, add a new record with the same date as in step 1.

When you open the Record Details screen to view the record added in step 2, the Related records field retrieves records with the matching date from the Lead Management app and lists them.