Why is Christianity often seen as part of a British Identity?

Because for most of the last 11 centuries, the various parts of Britain have been strongly Christian. This has also influenced the way the nation state has taken shape. Much of British identity is based around wars against Catholic powers, notably Spain and France, which has meant 'Protestantism = British" in the minds of many especially the Victorians. It is also why Ireland, which was always strongly Catholic, had a very different national identity and ultimately broke away from Britain (apart from Northern Ireland, where there was a Protestant majority in at least four of the six counties). The government of England in particular continues to have strong links to the Church of England, 25% of whose Bishops sit in the House of Lords, making the UK the only country other than the Islamic Republic of Iran to have clerics automatically appointed to its Parliament.

There are other reasons. The Church of England and Church of Scotland are organisations with a national reach - even today there are very few places in either country without a church or a kirk which acts as a focal point for the local community. In many places, they also manage the only recognised burial grounds for the area. In Wales, where religion was mostly around the Nonconformist churches (especially Baptists and Methodists) that is less prominent now, but much of Wales' musical and cultural heritage, which is crucial in keeping the Welsh language and thus Welsh identity going, derives from those areas and therefore it is still seen as part of Welsh identity.