Fake integers for which the Riemann hypothesis fails?

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.

Moving away from the Riemann hypothesis, some questions in additive prime number theory (e.g. twin primes conjecture or even Goldbach conjecture) are considerably more delicate than others (e.g. arithmetic progressions in primes or odd Goldbach conjecture) because in the former case it is easy to "redact" the primes by removing a small number of primes (e.g. all twin primes), reweight the surviving primes by the appropriate renormalising factor, and end up with a set of "fake" primes that is statistically indistinguishable from the original set of primes, except that a conjecture that was once believed to be true is now false. This all but rules out a large number of existing proof techniques for such problems, such as sieve theory or the circle method, unless these techniques are somehow combined with a new method that can distinguish the true primes from their fake counterparts. I discuss this in Section 2 of https://terrytao.wordpress.com/2012/05/20/heuristic-limitations-of-the-circle-method/ .

In a similar vein, one can replace the natural numbers by a fake version of the natural numbers in which products of an even number of primes are counted with double weight and products of an odd number of primes are counted with zero weight, or vice versa. Assuming a standard conjecture in analytic number theory (the Mobius pseudorandomness conjecture), these fake natural numbers are also statistically indistinguishable from the true natural numbers by all the tests for which we can hope to provably compute the statistics for the true natural numbers. On the other hand, any problem subject to the "parity barrier", such as the twin prime conjecture, can end up having the "wrong" answer when one replaces the natural numbers with this fake version (or some variant thereof), leading to the famous "parity problem". I discuss this for instance at https://terrytao.wordpress.com/2014/11/21/a-general-parity-problem-obstruction/

Also, in addition to Beurling's ideas, there is Landau's example of $\zeta(2s)\cdot \zeta(2s-1)$, which has Euler product, meromorphic continuation and functional equation, but no zeros at all on the critical line (by Hadamard and de la Vallee-Poussin). If one objects that this is artificial, a response might be that this is simply the globally split case of zeta function of a quaternion algebra, which, up to finitely-many Euler factors is again Landau's example. The finitely-many exceptional Euler factors product on-the-line zeros of $1-p^{s-1/2}$ for ramified primes. The collection of such zeros, for any finite (even-cardinality...) collection of ramified primes is $O(T)$ to height $T$, which is a negligible proportion of the $O(T\log T)$ zeros to that height.