I had always assumed these cards were picked up after the match, but on second thought I guess that explanation doesn't work very well. IIRC you don't need to fight the correct character at all to get more of the cards, so Reimu just "picking up" these watch cards for example, after a battle with Marisa, makes no sense to begin with. Huh.
Also Okuu getting 4 watches and them suddenly being available for Meiling to use also makes no sense (or get this, four total watches but both Okuu and Meiling have all four) but it's fine as a gameplay mechanics sort of thing.
Now I really must wonder what's up with these cards and where they come from.