It is possible (or even a core functionality) having data historicized within a classic data warehouse. Data will be added to the data warehouse over time and it is possible to move in time over the data.
If I just want to use the data lake and to have also data historicization for the business user, would this be possible? And if yes, how would a possible approach look like?
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Your classic data warehouse will bring data together, modelled with time series at the centre.
Data lakes hold the raw data in the original format, which typically will not be stored with time series in mind. You are able to store your data so that the time series and historical changes can be worked out, but a data lake will be missing the pre modelled, easily accessible time series aspect of a data warehouse.
Yes - you can do it. If you just do inserts of data then you will have, by default, a full history of all your data.
The possible approaches would be entirely dependent on the technology you were running to support your data lake, how you have structured your data in the data lake, the tools your business users were using to access the data, etc. So without much more information from you it's not possible to give you an answer - other than the generic "yes, it is possible to hold historic data in a data lake"