It can easily be interpreted by the common listener as sharing those two meanings.
But the common listener or native speaker normally pays little attention in practice to the niceties of grammar and specific meaning.
To avoid ambiguity in our communications, however, we must become more sensitively attuned than the ordinary unaware native speaker, and be able to recognise these differences in meaning and make perfectly certain what that small word "both" is referring to.
In your sentence "We both can read the book and watch the film", the "both" is placed most clearly in relation to "we" (however the ordinary unaware native speaker might interpret it), and not to the verbs "read" and "watch".
If you were to move the word "both" to an alternate place so your sentence became, "We can both read the book and watch the film", it seems to me that the ambiguity is magnified by uncertainty. The problem is compounded by having not only two objects (book and film) but also by having two verbs (read and watch) - as well as the subject "we".
One possible remedy which offers itself is to place a clear stress on the "and" which joins the two clause-objects to indicate that "both" relates to both "read the book" AND "watch the film". This ruse works easily when spoken. When written, we would need to rely on italicising or underlining or emboldening the "and" might do the trick.
The best remedy, in my opinion, would be to re-formulate the sentence so that we use just one verb in relation to both book and film. That strategy will removes the major obstacle to the achievement of unambiguous clarity.
Here are three possible results as illustration:
1. "We can enjoy both the book and the film."
2. "We both can enjoy the book and the film."
3. "We can both enjoy the book and the film."
In number 1, the word "both" relates clearly to book AND to film.
In number 2, the word "both" becomes ambiguous in interpretation once more because it appears to relate to the verb "can enjoy".
In number 3, the word "both" relates clearly to the subject "We".
I hope my comments help you, and do not cause confusion.
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