Hi, James –
while the name Gilbert is quite definitely of Germanic origin, that is not to say that it cannot be French, English or Scottish. Many French families have surnames that were originally Germanic, which is understandable because the French nation even takes its name from a Germanic tribe, the Franks.
(The nation now called French were Celtic-speaking Gauls who, as a consequence of Roman conquest, took to speaking the Romance tongue.)
The name arrived in the British Isles by way of the army that accompanied William of Normandy in 1066, but that army had various origins – some were actually Norman, others were Flemish (usually from families that had settled in Normandy), and yet others were from other parts.
What is important is that the name became English (in the sense of the English language, whether in England or in Scotland), and that many families that are now thoroughly Scottish settled north of the border during the Middle Ages, having come from England or further afield.
Which clan you belong to is a matter of your family’s history in Scotland, but don’t become over-anxious about links to England, France or Germany.
Regards,
Mike

PS: The modern French pronunciation of the name is Zhil-bear, but it could well have sounded much more like its English form in earlier times.