As it's just a private individual involved (rather than limited companies or unincorporated associations), there's no tax on either business, per se. The only tax payable is Income Tax, which is payable on an individual's net taxable income. (i.e. the sum of his/her profits minus their losses).
Irrespective of how many self-employed businesses someone is involved in, they can only submit ONE tax return, as there's only one taxable entity. (As stated above, it's the INDIVIDUAL who pays tax, NOT the businesses individually).
The person you refer to can either enter just one business name and description on his/her tax return (e.g. 'Freda Bloggs, general trading and property rentals') or two (e.g. 'Freda Bloggs, general trader' and 'F R Bloggs, t/a Grotty Homes, property rentals'). It won't make any difference anyway, as it's only the TOTAL figures that HMRC are interested in.