Yes, they have two date systems in common use. It's only the year that changes though. And there's no way to confuse the two, usually. If you write "2023" instead of "令5" it's pretty obvious. I suppose there is a potential for confusion if one just writes a two-digit year though.
Whenever their Emperor changes, the year starts with a new name(年号, which translates to name of year(s)). This time it's 令和(reiwa). Before that it was 平成(heisei). It is very commonly used.