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.

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

One way of making "fake integers" explicit is a Beurling generalized number system, which is the multiplicative semigroup $Z$ generated by 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 counting 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.

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