1) It looks like you must have use "day" as an input to groupsummary, which returns a categorical as the group values:
>> tt = timetable(datetime(2022,3,randi([1 3],10,1)),rand(10,1))
____________________ ________
01-Mar-2022 00:09:43 0.45054
01-Mar-2022 00:47:39 0.083821
02-Mar-2022 00:18:40 0.22898
03-Mar-2022 00:31:42 0.91334
03-Mar-2022 00:09:56 0.15238
01-Mar-2022 00:36:07 0.82582
02-Mar-2022 00:15:46 0.53834
02-Mar-2022 00:39:14 0.99613
01-Mar-2022 00:41:21 0.078176
02-Mar-2022 00:44:53 0.44268
>> groupsummary(tt,"Time","day","mean")
day_Time GroupCount mean_Var1
___________ __________ _________
Categorical is what those "convenience" flags like "day" do. But you don't have to use that. The simplest way to get the group values as datetimes is like this:
>> tt.Day = dateshift(tt.Time,'start','day')
____________________ ________ ___________
01-Mar-2022 00:09:43 0.45054 01-Mar-2022
01-Mar-2022 00:47:39 0.083821 01-Mar-2022
02-Mar-2022 00:18:40 0.22898 02-Mar-2022
03-Mar-2022 00:31:42 0.91334 03-Mar-2022
03-Mar-2022 00:09:56 0.15238 03-Mar-2022
01-Mar-2022 00:36:07 0.82582 01-Mar-2022
02-Mar-2022 00:15:46 0.53834 02-Mar-2022
02-Mar-2022 00:39:14 0.99613 02-Mar-2022
01-Mar-2022 00:41:21 0.078176 01-Mar-2022
02-Mar-2022 00:44:53 0.44268 02-Mar-2022
>> groupsummary(tt,"Day","mean")
___________ __________ _________
2) I'm confused. None of the "dates" in day_time_date appear in time_date.
Your question might be, "which rows of time_date correspond to each row of day_time_date?" That's easy, once you have tt.Day:
ismember(tt.Day,day_time_date)