"Dame la mano" would mean "give me the hand". Shouldn't it be better to say "give me your hand" by saying "dame tu mano"?
For reference this is the entire phrase coming from spanishdict: "Dame la mano, no vayas a perderte entre tanta gente."
"dar la mano" is an idiom. Although we can hear:
it's more idiomatic, as well as more logical, to say:
In Spanish, we tend to use articles rather than possessives with parts of the body in direct object position. This is not only the case with reflexive actions but also with actions performed on others. We prefer to use a pronoun to indicate the recipient of the action and an article rather than the possesive. (Notice the ambiguity with the English possessive in the sentences between parentheses below.)
We thus say:
Le lavó la cara (He/She washed his/her (somebody else's) face).
Se tocó la frente. (He/She touched his/her (own) forehead.)
Le tocó la frente. (He/She touched his/her (somebody else's) forehead.)
Se tapó los ojos. (He/She covered his/her (own) eyes.)
Le tapó los ojos. (He/She covered his/her (somebody else's) eyes.)
Se cortó el pelo. (He/She cut his/her (own) hair -- or had it cut.)
We only don't use a pronoun when our hands or fingers do not touch the part of the body involved. In these cases, English uses a possessive, while Spanish requires an article: