Nah, when I played the game it didn't make sense. But if Trish is alive and well in DMC2 the only things I could come up with considering the coin are those:
1) She left Dante for an unknown reason so that's why Dante has the coin as a mean to remember her, that's why he is so serious etc.
2) Dante went to that far away island in DMC2 without Trish since he felt it would be so easy that his powers are more than enough or sth. If he was mistaken he would contact her through the coin, that's why he left the coin with Lucia so that the contact could be established since if he went in the Demon World it would be difficult for her to find him.
But really, although Trish IS my favourite female DMC character there, why we didn't see her in DMC2 except for as a playable character?
I hope and believe that DMC4 will have a good explanation though^_^