After a research, it turns out that the issue is not reproducible with JAWS 18 (18.0.4350) or JAWS 2018 (build 1710.42) and IE 11.726.15063.0, but can be observed in IE 11.1770.14393.0.
As the implementation is based on the ARIA spec and the problem can be reproduced in a specific version of IE, either disable the EnableAriaSupport property or use the workaround of setting the role="presentation" attribute:


<script>
    function dateInputLoad(dateInput, args) {
        var calendarInputWarapper = document.getElementById(dateInput.get_id() + "_wrapper");
        calendarInputWarapper.removeAttribute("role");
        calendarInputWarapper.setAttribute("role", "presentation");
    }
</script>
 
<telerik:RadDatePicker RenderMode="Lightweight" ID="RadDatePicker1" runat="server" EnableAriaSupport="true">
    <DateInput runat="server" ClientEvents-OnLoad="dateInputLoad"></DateInput>
</telerik:RadDatePicker>