There is no statutory provision prohibiting a child from owning shares. At common law a child will not be bound by a contract to buy shares as they are not 'necessaries'. That may make it difficult to enforce payment for the shares against a minor. Some companies will not accept shareholders under the age of 18 years by provision in their articles or terms of issue.
In many family owned companies, shares are allotted to children as a means of providing them with capital assets which may be expected to increase in value as part of longer term inheritance and capital gains tax planning. In many such cases, different classes of shares are set up so that the child may, for example, have non-voting shares, or so that the directors can pay different rates of dividends on the various classes, giving flexibility as to how the company's profits are distributed. Such schemes should only be set up with professional tax advice, and great care must be taken to have the share classes set up properly.
Public companies often provide that minors may not hold their shares. Such shares are often held by parents or grandparents etc as trustees for children, or alternatively some form of investment trust is used.
Note that there is a restriction on children being directors of companies. A director must be over the age of 16.