However, some theoretical papers do this too.
There are several reasons why authors may do this.
Laziness. It can be hard work and confusing to work out the actual physical units for some theoretical calculations.
Complexity. For experiments it can be extremely difficult to normalise and calibrate some detectors.
Uncertainty and embarrassment. Parameters such as detector efficiency, sample thickness,
geometric corrections, solution concentration can involve large uncertainties so the horizontal scale may be unknown by as much as an order of magnitude.
But I think these uncertainties should be reported because they present a challenge for improvement.
I realise that the measured "units" may be detector dependent and not particularly useful to experimenters in different labs. For example, the units may be "number of clicks per second in homemade photodetector 3 in Professor Smith's lab".
But I still think those details should be reported.
1. It belies the fundamental fact that any physical quantity does have actual units.
2. For experiments it means that the reported measurements are not reproducible.
i.e. they cannot be checked. The shape of the spectrum can be checked but not the magnitude.
3. This practice limits the comparison of theory and experiment.
The absolute intensity [spectral weight] of a spectrum will be predicted by theory. It is important to test this experimentally. Just because a theoretical calculation gives the correct spectrum does not mean it is correct.
Absolute units allow one to test sum rules.
[Aside: I have a prejudice/suspicion that matrix elements may be generally more theory sensitive than energies. For example, in quantum many-body theory it is possible to get a very accurate ground state energy with a variational wave function that has a small overlap with the true ground state.]
Another example is the temperature dependence of transport properties.
Sometimes resistivity is reported in arbitrary units or the resistance [rather than resistivity is reported].
Simple theories can sometimes get the temperature dependence of the properties such as resistivity and thermopower correct but the absolute magnitude can be off by orders of magnitude.
For a nice example of people working very hard to normalise spectra, test sum rules, and get physical insight, see the paper
Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain.
A less impressive example [that I was involved in] is in the paper Transition dipole strength of eumelanin.
So, do the hard yards. Don't use "arbitrary units".
If you referee a paper that does, request the authors to do better.