A tea-poy has very little to do with tea. It was originally the name for an Indian three-footed, i.e., a tī-pāī table which could be used for many things including serving tea.
The Hindi/Urdu तिपाई itself comes from the Sanskrit त्रि (trí, “three”) + पाद (foot). A charpoy is similarly, a four (char) footed (pai) bed.
Connotations
Indian connection