Generally pseudo random number generators don’t work like that. That would indicate that the sequence would repeat after .Machine$integer.max steps. The better prngs have much more states than .Machine$integer.max and integer seeds represent only small portion of those states and those states are very far away. Stan uses ecuyer1988, which has length of cycle 2^{61} and .Machine$integer.max is 2^{31}, so there are plenty of pseudo random numbers between each seed.
Mathworks did some carefully analysis years ago choosing their default prng, which is now Mersenne twister. Mersenne twister mt19937 is also what boost recommends as the first choice. It has length of cycle 2^{19937}-1. Use of ecuyer1988 in Stan has been questioned before Quality of Pseudo-Random Number Generators, but for some reason only other than boost library solutions (e.g. mt19937) were considered upgrade our PRNG · Issue #403 · stan-dev/stan · GitHub
mt19937 would be also 13 times faster than ecuyer1988, but I guess speed has not been a problem.