。
She has as many books as I do.
Literal
She [topic-は] me [as-と] same-amount books [object-を] possesses.
The equality construction X と同じくらい Y ('as Y as X') uses 同じ ('same') with くらい ('approximately, about') and the comitative と ('with / as'). Literally 'about-the-same-as me, possesses books' — the comparison standard comes first, then the dimension of comparison. 持っている marks possession: the stative ~ている of 持つ ('to hold / possess'), since 'having' is a continuing state, not an action you keep doing.