Horn is made of keratin, which is the same stuff that your finger and toes nails are made of. This does not survive well in the archaeological record as it rots very quickly. As such it could be argued that the number of horn finds are not entirely representative of the volume in actual use.
Horn becomes pliable when heated in oil and can be shaped over formers. Subsequently dishes and bowls can be formed, however, the technology to keep the horn within the required temperature range to do this without damaging it is modern.
Small horn eating size spoons are similarly questionable on authenticity grounds, but they are permitted on the basis that many people have difficulty eating with a wooden spoon.
One of the York Mystery Plays (from the later medieval period) includes the shepherds bringing the infant Jesus:
There is no evidence of bowls made from horn, so the use of large formed bowls, dishes and large spoons etc is to be discouraged.
Horns are also commercially available made into cups or tankards with side handles. Unfortunately these are re-enactorisms - you should use pottery or wooden cups instead.