Are there any astronomical phenomena that could emit strong radio waves with multiples of a discrete frequency? In the New Scientist article Is this ET? Mystery of strange radio bursts from space, it is reported that several times since 2001, astronomers have detected fast radio bursts that seem to have a frequency of multiples of 187.5.  In the article, the theory that it could be a pulsar is discounted.

Are there any astronomical phenomena that could emit strong radio waves with multiples of a discrete frequency?
I think you've misunderstood the article - the quantity that seemed to be occurring at integer multiples of some number isn't the frequency of the radio emission but rather the dispersion measure (DM) of the source. As photons travel through the interstellar medium, interactions with free electrons mean that lower frequency photons take longer to reach the observer than higher frequency photons. The time delay of between photons with frequency $\nu_1$ and $\nu_2$ is
$$t_1-t_2=4.15\left(\frac{\text{DM}}{\text{pc cm}^{-3}}\right)\left[\left(\frac{\nu_1}{\text{GHz}}\right)^{-2}-\left(\frac{\nu_2}{\text{GHz}}\right)^{-2}\right]\;\text{ms}$$
with the dispersion measure defined as the integral of the free electron number density $n_e$ over the path between the source and the observer:
$$\text{DM}\equiv\int n_e\;d\text{l}$$
In a sense, the dispersion measure is a distance proxy. If we had some uniform $n_e$, then the DM would scale perfectly linearly with the distance to a source. That said, the Galactic electron density is not uniform in all directions; it's obviously much higher in the disk, meaning that a source inside the disk some distance from us will have a lower DM than a source in the halo the same distance away. Note that the DM isn't an intrinsic property of the source - it depends on the observer's location.
The point of the paper cited in the article (Hippke et al. 2015) is that the dispersion measures of the 11 fast radio bursts (FRBs) appeared to fall near integer multiples of 187.5 pc cm$^{-3}$. There's certainly variation, although I don't know how the deviations compare to the measurement uncertainties. The authors argue that this could be explained if FRBs are actually artificial - in other words, if there's some sort of human-made source emitting at different frequencies with each frequency emitted at a different time mimicking the effects of interstellar dispersion.
Our knowledge of FRBs has since grown as the sample size has expanded dramatically, and most known DMs vary quite a bit from the trend. I don't know if anyone has looked at this small sample again, but at this point, the extremely broad consensus is that FRBs are astrophysical (mostly extragalactic) sources. A broad range of mechanisms have been proposed over the years, with some since ruled out and others supported by further observations.
A note on pulsars: The article isn't stating the pulsars aren't FRB sources - the sample size back then was far too small to rule them out - but rather that fast radio bursts differ significantly from typical pulsar emission, which tends to be uniform. There are still viable FRB mechanisms involving pulsars, such as giant pulses of young pulsars, and neutron stars in general play a role in some currently-favored theories, such as magnetar flares.
All that said, I think it's worth answering the title question. It is indeed true that some radio sources emit at integer multiples of some frequency - namely, spectral lines from rotational transitions in molecules. In a molecule with moment of inertia $I$, the energy of the state with the angular momentum number $J$ is
$$E_{\text{rot}}=\left(\frac{\hbar^2}{2I}\right)J(J+1),\quad J=0,1,2,\cdots$$
A transition from $J$ to $J-1$ then releases an energy
$$\Delta E_{\text{rot}}=\frac{\hbar^2J}{I}$$
and leads to the emission of a photon with frequency
$$\nu=\frac{\hbar J}{2\pi I}$$
As a notable example, the $\text{CO}$ molecule$^{\dagger}$ emits a photon at $\nu\approx115\;\text{GHz}$ during the $J=1\to0$ transition, at $\nu\approx230\;\text{GHz}$ during the $J=2\to1$ transition, and so on. Keep in mind, though, that fast radio bursts are broadband sources, and while spectral radio lines will exhibit some broadening through certain mechanisms, it's simply not comparable - the emission profiles are completely different.

$^{\dagger}$Specifically, the common isotope $^{12}\text{C}^{16}\text{O}$. Other isotopes like $^{13}\text{C}^{16}\text{O}$ have corresponding lines at slightly different frequencies.