Analog filters have classifications too.
Active filters such as Gm-C, SC, Cherbyshev, Butterworth, KRC, Sallen-Key, Unity and etc.
Passive filters made from RLC components.
SC is the fundamental principle of discrete-time components. FIR and IIR are delay-based filters, i.e. casual systems that have memory of time-interleaved inputs or data. The latches or delay elements are implemented using SCs.
Digital filters that you know such as FIR and IIR filters are synchronous discrete-time filters and these are classified in nature of the physical structure in which these filters operate. There are asynchronous filters that are digital but perhaps you are not aware of.
Analog filters are classified in so many ways that simply structural differences are not the only criterion to differentiate them. Analog filters are also differentiated in terms of their phase margin, frequency response, Q factor, order, -3 dB bandwidth, gain, and sometimes poles and zeros are used to determine the characteristics.
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Analog Devices Inc (Ireland)