Is this the right way? A simple signed letter/email?
I’d do it by letter rather than e-mail and keep a copy for yourself just in case.
May depend on how big your place is and what it may say in your rental contract about the number of people that can live there.
But honestly, it might be best to make a written sublet contract With this it is known how much shall contribute (you can still waive it) and more importantly how you can kick out your brother.
Would something as simple as the following be ok?
" I [my name], tenant of the apartment at [street], hereby confirm that [bro's name] will be staying in my apartment and this should be regarded as his official residential address in Switzerland until further notice. "
@Medea Fleecestealer: I couldn't find any clause in the contract limiting the number of tenants, but it might be in the small letters. Good point, thanks!
Is the brother a (sub)renter though? I think he’s a guest. OP appears to concur, calls himself host.
Imagine his stay lasted two weeks, I figure that would be a clear case for a guest. I think the nature of the stay doesn’t change if it’s “a couple months”, the intended duration is short in both cases even if there’s no predefined end date.
Nonetheless it won’t hurt to have a written agreement. Not because of distrustm, quite the contrary, but to avoid problems and protect the relationship - “harte Geschäfte, gute Freunde”. Cleaning the bathroom, kitchen/ware, and apartment may be worth talking about, washing as well, mention that rent and other costs are zero. Who pays if stuff gets broken, keep in mind that the 3rd party liability insurance (do remember to take one) doesn’t pay for family members living in the same household. Perhaps also consider what’s mentioned in the template for subrent by the Mieterverband.
So, I emailed them, they sent me back a simple 1-page sublet form and all good. The form had a start & end date of the subletting, I left the end date open (no clue how long it will take for him to find a new apartment in the current situation), and they told me to let them know if he stays more than 6 months.