Not directly, no - but I would guess that major donations to museums in the UK are pretty thin of the ground too. Some solicit 'donations' from the visiting public, but from what is visible to the general visitor this produces only a tiny part of their operating requirements. The major difference is that most city and National museums in the UK are funded out of public budget and enjoy very different tax advantages; whereas most Zoos are subjected to pretty punitive levels of taxation.
Just because I happened to attend a talk at work yesterday about museum funding (disclosure: I have worked in UK museums for over a decade), central government funding of national museums, although significant, now amounts to no more than 40% annual income for any single national museum since the funding cuts after the recession (before the recession, central government funding could be as high as 80% annual income). At present, national museums will be generating 60% of their own annual income, including through donations, bequests, Friends/Membership schemes, charges for exhibitions (both on-site and touring), shop sales etc. As central government funding will undoubtedly keep falling, the percentage of self-generated funds will continue to increase.
Compared to the regional and city museums, nationals have it easy. Regional and city museums have suffered an even bigger drops in government or council funding, and don't attract the same level of attention as regards donations or benefactors. They used to receive a good boost from the Renaissance in the Regions funding programme for regional museums, but this was stopped around 8 years ago, and no new source of funding has materialised.
I am less certain of the tax status for museums, but as most are registered charities, they are probably taxed similarly along those lines.