Replying to my own question, a couple of points:
- Tener is not used in the same way as English to have when it means "to receive" (an item) or "to consume" (a drink, a bite of something).
- Tener cannot be used to ask for something as in the English formula Can I have...? Neither can recibir "to receive, to get, to obtain".
The Spanish structure used to ask for an item (to keep, use or consume) usually involves the verb dar "to give". That is, it involves the giver as well as the receiver:
¿Me darías un vaso de agua?
which literally means: "Would you give me a glass of water?", although more naturally you would say, "May I have a glass of water?".
Instead of dar there are several verbs involving taking an object and passing it to someone, such as pasar, alcanzar, etc.
¿Me pasarías ese informe?
(lit.: "Would you pass me that report?",
also "Can I have that report?")
The verb can be used in the indicative mood, or in the conditional mood (as in my examples) for a softer, indirect approach, or combined with poder (even more indirect):
¿Me podrías alcanzar aquel libro?
"Could you (reach out and) pass me that book?"