In practice, yes, to add up TSS for a week’s or a month’s training, you just add up the TSS of the individual sessions.
As you suggest, the cumulative TSS of two workouts, recorded separately but with the second performed immediately after the first, may not be the same as the TSS if you recorded the two workouts as one event.
TSS is a approximation useful for tracking your work. It’s more relevant than distance or time or calories, for instance. But it has limitations.
TrainingPeaks is the home of TSS:
The formula for TSS is:
TSS = (sec x NP® x IF®)/(FTP x 3600) x 100
“sec” is duration of the workout in seconds,
“NP” is Normalized Power® (don’t worry about this for now),
“IF” is Intensity Factor® (a percentage of your FTP; in other words how intense the effort was),
“FTP” is Functional Threshold Power (your best average power for a one-hour race or test),
and “3600” is the number of seconds in an hour.