There is no equivalent. If you allow electromechanical, then the equivalent of a uniselector is a uniselector. There simply is no other component with sequential operation and mechanical contacts. Perhaps there are units with extra stuff tacked on, and sold under different names (e.g., perhaps someone makes a manual selector switch that can also be remotely advanced), but in as much as the basic functionality is there, I mean, it's just defining a tautology.
It's a complicated mechanism; computers have been built using them, plus relatively fewer relays in addition. That is to say, uniselectors save a fair number of relays compared to an all-relay design (where "relay" means ordinary say SPST to 4PDT types, latching or regular), and so we should expect to need a fair number of relays to replace one in general. They're logically powerful devices.
If you disallow electromechanical, then there is no exact substitute, fairly obviously, but even just in signal terms. Mechanical contacts are unmatched in power handling, bandwidth and on/off ratio. Transistors work in limited circumstances: there are SSRs of various types that are capable of significant power at low frequencies (usually mains frequency or including DC), but are limited in Rds(on) (or similar metrics); or at high frequencies, but are limited in signal level (e.g. RF and logic switches in the say 0dBm to 5V range). Even these are limited to the low GHz, while a good RF relay is limited only by its geometry / physical size. Likewise, at DC, an on/off ratio of some billions is typical for a transistor, but thousands to millions of times higher for mechanical switches.
Conversely, there are generally-undesirable traits that mechanical contact have, that transistors don't: contact bounce; slow operation; contact resistance; breakdown (arcing); and the need for wiping (mechanical motion, often while carrying load current), or manual cleaning.
