No, you can keep pet cremains indefinitely, since they are not human remains. However, if you’re planning on staying in the same locality (e.g., the same city,) and if there is a pet cemetery in the area, you might want to check with them what their policies are. They might be a lot more financially reasonable than you’d think. There is a pet cemetery near us, and you can get a plot large enough to bury a horse (intact,) for about $200 for perpetual care (grass mowing, keeping grass trimmed back around headstones, that sort of thing.)
When I bought the plot, they told me that up to twenty cremated household pets could be buried in it. Each time they have to open the plot, they charge $50 for that, and there is a smallish marker with “beloved friends of (our family name)” on it. I’m pretty confident that the pet cemetery will be there for at least as long as I’d be around to care one way or another, they’ve been in business since the 1890s and the land is permanently deeded by the county as being a cemetery, meaning it can’t be used for anything else.