。
She has as many stamps as I do.
Literal
She [topic-は] me [as-と] same-amount stamps [object-を] possesses.
An equal-comparison frame XはYと同じくらい~ here applied to a collection of stamps. 切手 ('postage stamp') is one of the classic Japanese collectibles — 切手収集 ('stamp collecting,' philately) was hugely popular in the postwar decades, and Japan Post still issues commemorative sheets several times a year. The ~ている of 持つ keeps possession in stative form.