Case could be made that BB stands for ball bearing which is metal and not rubber.

Pellet is a more general term tho.

The term BB originated from the nomenclature of the size of lead shots used in a
smoothbore shotgun. Size "BB" shots were nominally 0.180 in (4.6 mm), but tended
to vary considerably in size because of the loose tolerances in shotshells. The largest
shotgun pellet commonly used was named 00 or double ought and was used for
hunting deer and thus called buckshot, while the smaller BB-sized shot was typically
used to shoot small/medium-sized game birds and called birdshot.

A BB gun is a type of air gun designed to shoot metallic spherical projectiles called
BBs (not to be confused with similar-looking bearing balls), which are approximately
the same size as BB-size lead birdshot used in shotguns

The term "BB gun" is frequently used incorrectly to describe airsoft guns, which shoot
plastic pellets (also often referred to as "BBs") that are larger (usually 6 mm or 0.24 in
in diameter) but much less dense than metal BBs,[1] and have significantly lower ballistic
performance. The term is also sometimes used to describe a pellet gun, which shoots
diabolo-shaped (not spherical) lead projectiles at higher power and velocity. Although
some BB guns can also shoot pellets, the reverse situation is not true: steel BB balls
have greater stiffness and are not meant to be shot from pellet guns, whose barrels are
typically rifled and thus can get stuck (similar to a squib load in firearms) and lead to a
damage or mechanical failure within the pellet gun.