Does an audio equalizer consist of a perfect-reconstruction filterbank?
A filter bank system is generally composed of:


an analysis filter bank, that split an input signal into components with filters and may reduce their rate
a synthesis filter bank (FB), that takes the components, potentially increase their rate, and feed them into filter elements 


so that the novel outputs can be combined into a signal. It is perfect reconstruction (PR) when, for every input, the output is a potentially delayed or scaled version of the input. Filters can be nonlinear, the rate reduction can be different for each component. Min-Max or morphological wavelets are examples of the above. There are even hybrid filter banks, where the analysis (A) or synthesis (S) FB are either analog or digital. Their design is quite involved, so often A and S are linear, with upsampling and downsampling operators, and an output obtained by summation, as explained by @Marcus Müller. However, the filters may have different bandwidths and the subsampling can be different for each filter. 

Many signal processing systems, such as (most) equalizers, are made of an analysis FB, a processing unit (often nonlinear) and a synthesis FB (which could be a mere summation). The non-perfection arises, for PR FB, because of the middle filtering component. 

Assuming an equalizer has linear A and S filters, the non PR aspect may arise from two aspects. First, what are the filters? It seems to me that the traditional filters (first or second order) do not allow easy PR. However, with sufficient frequency overlap, they together can form a near PR filter banks, with an error below the hearing threshold.
The second aspect is the processing. If you suspect saturation, or nonlinear phase alteration, then PR is not achieved in  general.

Phase distortion is sometimes considered harmless for audio in some   textbooks. However, picky sound engineers are cautious about audible phase distortion. The article Plugging into EQ discusses those issue, around FIR or IIR filters. The first ones can have linear phase, but have less sharp frequency transitions, and require a handful of computations. The second ones do not have linear phase in general. It also provides a comparative evaluation of several EQ.

So, as Markus, I'll say no in general, with a modest yes (in cases) if below the hearing sensitivity.