[Dougie Jones]

This works, also with repeated id values

clear
input float(id otherid)
101 102
102 104
103 101
103 102
end
gen n=_n
rename (id otherid) (idorig idother)
reshape long id, i(n) j(idtype) string
egen newid = group(id)
reshape wide id newid, i(n) j(idtype) string

This works perfectly fine and is, in my opinion, the most elegant solution to my issue. Thank you all!