The character 分 has two different readings. As fen1, it has a range of meanings. As fen4, it can mean a role or part played by a person, a more general part or portion of something, or a component. Fen4 can also be written 份, and dictionaries I consulted from both Taiwan and the mainland don’t seem to differ here.
The Far East Chinese-English Dictionary, pretty much the standard bilingual dictionary in Taiwan, has the following: 身分證 or 身份證 shēnfènzhèng ‘ID card’.
The Chinese-English Dictionary from the Foreign Language Teaching and Research Press is a mainland publication. It has 身分证 or 身份证 shēnfenzhèng ‘ID card’. This dictionary does one thing a bit differently – it divides the meanings of fen4 into those which can be written with either character and those which use only 分. In the latter category is 分外 ‘especially’. But this doesn’t affect the term for ID card.
There may be some cross-strait difference in actual usage between分 and 份 in this word, but as I have shown here, according to dictionaries both forms are acceptable wherever you are!