Can this european sauna heater be installed in the US?
Yes you can, provided your AHJ is OK with the unit, but you should be treating it as a single phase unit -- you'll also need to take precautions with wiring methods

The instructions you linked provide for a single phase, 230V, 48A connection as well as the 3-phase, 230/400V, 16A connection you discuss in your post -- the single phase connection is what you want, since the type of 3 phase power this requires (400Y/230V) doesn't exist on this side of the pond.  Thankfully though, mains frequency doesn't matter to heaters, and a sauna heater like this does not need to be listed, just signed off on (approved) by your AHJ -- sauna-type heaters are not included in the NEC 424.6 list of space-heating equipment that requires listing.

The major caveat though is that you'll need to use a wiring method that's rated for both the high temperatures in the sauna room and a damp (at minimum) location.  While the type of cordage the installation instructions call out is not rated for North American usage, an equivalent cordage type is available through some online suppliers or electrical supply houses -- industrial-grade multiconductor type W portable power cable.  You'll want a 4/3 type W cable for this (hot, hot, ground) and to route it from the heater to a weatherproof (NEMA 3R or 4X/12) junction box located in the sauna room somewhere with an appropriately sealed gland/strain-relief where the cord enters it (at the bottom, too).  The use of cordage here is permitted by NEC 400.10(8) due to the equipment in question being identified for flexible cord connection, by the way -- in fact, I suspect we're dealing with the weird case of a fixed piece of equipment that is identified only for flexible cord connection, not for the use of a Chapter 3 wiring method directly to the device. (Which is unfortunate, if you ask me -- the proper way to do this would be a rigid conduit with type SA wire all the way to the device, but that's out of the question as the install instructions don't support that.)

From this box, you'll need to run a rigid metal conduit out of the sauna room, with properly sized individual XHHW-2 (note: NOT THHN/THWN-2 -- PVC wire insulation does not hold up well to extended exposure to wet heat -- XHHW-2 is a bare minimum for this application, but I'd use type SA wire if I can get it as it's fully rated for sauna conditions) conductors inside it.  (The manufacturer's instructions call out the metric equivalent of 6AWG wire, but if you were to apply full NEC derates to the XHHW-2 conductors, you'd need 1AWG copper or 1/0AWG aluminum instead.)  Once the conduit run exits the sauna, you can then transition to a saner wiring method for the homerun, such as 6/3 NM with the neutral capped off at the end of the homerun, at the box for the contactor that controls this thing, based on whatever you are using for a switch or thermostat.

With this, you'll want to use a 60A, 2-pole breaker (48A * 1.25 for a continuous load) of the type that's normally used in your electrical panel.  (The proper application of Eurostyle MCBs in North American electrical systems is a subject far beyond the scope of this question, and a North American panelboard-type breaker can protect a dumb heater just fine, thank you very much.)