One way of making "fake integers" explicit is a *Beurling generalized number system*, which is the multiplicative semigroup $Z$ generated a (multi)set $P$ of real numbers exceeding $1$; lots of research has been done on the relationship between the counting function of $P$ (the *Beurling primes*) and the conuting function of $Z$ itself. In this context, it is certainly known that the Riemann hypothesis can fail; see for example [this paper of Diamond, Montgomery, and Vorhauer][1].

If this is not "similar enough to the integers", then I think you should more clearly define what you mean by that phrase.


  [1]: https://deepblue.lib.umich.edu/bitstream/handle/2027.42/46253/208_2005_Article_638.pdf?sequence=1&isAllowed=y