Would a higher air pressure on the ISS or elsewhere make it easier to "swim" in microgravity?

There is some misconception involved in the phrasing of the question. Take a look at the ideal gas law:

$$frac{pV}{nT}=rm constant$$

$p$: pressure; $V$: volume; $n$ amount of substance ("mass" of the gas); $T$: temperature

What you need to do in order to increase the swimability is to increase the density, which is the ratio $frac{n}{V}$. Assuming the volume $V$ of the space station's modules remain constant, you'd need to increase $n$ by pumping more atmospheric gas into the station.

By that law, the pressure $p$ would inevitably rise, leading to problems stated in @Uwe's answer. Although our atmosphere is not ideal but a real gas, one can conclude:

Yes, but one would have to manage Nitrogen narcosis as discussed in @Uwe's answer.

If you insisted on increasing the pressure without increasing the mass, you could change the temperature. But this is just a theoretical answer, as a temperature of around 1500 K is necessary to reach a pressure of 5 atm. In such an environment, the astronauts would not be able to do anything but to evaporate.

To improve swimmability, we need to increase gas density, not gas pressure - although both are related, it would be ideal to increase the former without increasing the latter.

Density of fluids can be increased by solids in suspension, as can be shown by hot pyroclastic flows denser than colder clean air. In Earth solids in suspension tend to settle due to gravity, but in space anything floating in the station atmosphere keeps floating there. Then, we can suspend in air a lot of mass and keep pieces large enough to not interfere with breathing. Therefore, the solution is:

The big micro-gravity ball pit

We just need to let some thousands of solid rubber balls floating in the station. When swimming, astronauts will trow back a large mass of balls with a little mass of air.

To optimize the system, balls must be large enough not to be swallowed, as massive as possible, not very hard to avoid hitting hard the astronauts and elastic, so they bounce on the walls instead of setting against them. Solid rubber balls a few centimetres of diameter seem a good trade-off between those requirements.

Of course, with a couple of balls for litre of air visibility will be highly impaired, but that's just a secondary effect to bear on.

Would a higher air pressure on the ISS or elsewhere make it easier to “swim” in microgravity?

Yes!

But what's really important is the density, so instead of pressuring "normal air" you can just make a denser atmospheric mixture and keep the pressure the same.

This answer says

If you want the air to be 5 times easier to swim, you can just replace the nitrogen with xenon and increase the density without increasing pressure.

and while it is pointed out that Xenon is expensive and has a narcotic effect (this guy complains of tingly fingers from Krypton before breathing Xenon), so what about this?

Wikipedia's Sulfur hexafluoride says:

Sulfur hexafluoride (SF₆) is an inorganic, colorless, odorless, non-flammable, non-toxic but extremely potent greenhouse gas, and an excellent electrical insulator.

Consider using a normoxic mixture (normal oxygen fraction of about 21 %) of SF₆ for a while, but not permanently!

From Effects of Sulphur Hexafluoride on Psychomotor Performance:

The narcotic influence of sulphur hexafluoride on mental and psychomotor performance has been studied in 9 subjects at normal atmospheric pressure. Control experiments were performed with air and with nitrous oxide. Psychomotor, perceptual and cognitive abilities were assessed using a computerized test battery. Subjects were exposed to air and six different normoxic gas mixtures: 13, 26, and 39% N2O, and 39, 59, and 79% SF6. Significant performance impairments were found with 13% N2O and gradual further impairment with 26, and 39% N2O. During exposure to 39, 59, and 79% SF6 over-all performance was impaired by 5, 10, and 18%, respectively. Impairment was significant with 59 and 79% SF6. The results indicate that the relative narcotic potency of SF6: N2O is about 1:4 in humans. It is concluded that a normoxic SF6-O2 mixture can be inhaled for lung function studies without any harmful effects and that the short-lasting narcotic effect, although detectable with a test battery, would not impair the ability of the subject to perform simple breathing procedures.

Also see Relative narcotic potency and mode of action of sulfur hexafluoride and nitrogen in humans

Physics

In microgravity the ability "swim" in an atmosphere comes from the aerodynamic drag force produced on the astronauts fast-moving arms which is approximately

$$F_D = frac{1}{2} rho v^2 C_D A$$

where $rho$ is the density of the atmosphere, $v$ is velocity, $C_D$ is the drag coefficient which contains all of the fluid dynamics but is usually somewhere between 0.5 and 1, and $A$ is the area considered.

Since arms pivot at the shoulder each part moves at a different speeds, let's say an area of 0.01 m^2 does most of the work, and it moves at about half of the world's record speed for a thrown ball of 22 m/s (from this answer to How hard do you have to throw something off the ISS to make it deorbit?). The density of a standard atmosphere is about 1.225 km/m^3 and let's use $C_D$ of 0.5 for a non-optimal flailing arm.

That makes the drag force about 1.5 Newtons! Assuming the double arm swings are underhand to keep the force near the center of mass, a total of 3 Newtons over a 50 cm arc. With work equal to force times distance, that's 1.5 Joules of kinetic energy.

The "delta-v" the astronaut receives from each double-armed underhanded flail is then

$$Delta v = sqrt{2E/m}$$

or about 0.2 m/sec. That seems much faster than what a single flail gives the astronaut in the videos (Astronaut gets stuck in the Kibo ISS model and It can be difficult to remove oneself from the Kibo ISS module) but it's the right order of magnitude.

And a factor of 4 if not 5 in density from an ~79% SF6 atmosphere would be a big boost!

If you want the air to be 5 times easier to swim, you can just replace the nitrogen with xenon and increase the density without increasing pressure.

Partial answer to "Is it a proposal space agencies should consider?"

Unlikely. Increasing the differential pressure by a factor of 5 would mean that the modules would have to be quite a bit stronger and therefore presumably costlier and/or heavier. (As pointed out in this other answer)

If getting marooned in midair is a constant problem (AFAIK it isn't) ¹ a much cheaper and lighter solution would be to string tethers down the long axes of the modules. Swimming in the air is not a design requirement.

¹ This answer quotes early ISS astronaut Dan Barry as saying "It's not easy to get stranded - I had to have my friends help me get perfectly still."

The astronauts would get nitrogen narcosis even worse than in 40 m deep water breathing air. In both cases the gas pressure is 5 bar, but under water the partial pressure of nitrogen is 3.95 bar but in the spaceship 4.79 bar. This is equivalent to about 50 m deep water breathing air. See Wikipedia for signs and symptoms of the narcosis. These symptoms would endanger the life of a diver or astronaut.

But the spaceship would get too heavy anyway when built for 5 instead of 1 bar.

To avoid decompression sickness during an EVA, a partial pressure of nitrogen of 4.79 bar can't be used. A space suit pressurized to 5 bar is totally useless, so pure oxygen with about 0.3 to 0.4 bar is used to keep the suit flexible. A very long decompression procedure (several days) would be needed to avoid decompression sickness during transfer from 5 bar to only 0.4 bar.

So to avoid all these problems, high pressure swimming is impossible.