That scene's told more or less from the point of view of Frodo, who, being of a wealthy, scholarly (Bilbo) and somewhat eccentric (Bilbo again) family could have been at school, and could have thought that Sam looked 'as if' he'd been there too-- although Frodo well knew that Sam wasn't. I'm not sure if hobbits had schools, but they probably did have small private ones, which ended about late adolescence. They certainly didn't have later secondary schools (for ex., high schools in the U.S.) or universities.
The other answer is that JRRT's style in the early part of the book was more informal and colloquial, with insouciant out-of-period references. So when he says 'school' he's talking to a reader much like him who had been to school and not worrying if that's the correct point of view for Sam's or Frodo's experience.
Sam definitely did not go to school: he was of too poor and humble a family and it was an act of generosity and a blow against the local social strata that Bilbo taught him his letters and some elvish poetry. That's one reason Sam was so determined to take care of Frodo; he was very grateful to Bilbo for seeing more in him than a pair of hands and a strong back. No one in the gaffer's generation thought he had a mind, so his intelligence was spent in crusty wit and crankiness-- the only reason it was different for Sam is that Bilbo took a generous and RESPECTFUL interest in him. Bilbo was always very casual about it, but in his own way, he came back from the Misty Mountains a radicalized hobbit-- which we can attribute to the effect of Gandalf's company and the experience of watching the highborn and 'noble' get it all disasterously wrong (Thranduil and Thorin).
[ September 25, 2002: Message edited by: Nar ]
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