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In OpenTTD, it's possible to build multi-platform train stations side-by-side (pictured below as Parnpool Mines), and they would work as expected, allowing two trains to load/unload at that station at the same time.
However, sometimes due to space constraints it would be more desirable to to build the two platforms lengthwise as shown below in Parnpool West. The signal-station combination pictured below, however, does not work, and I have not found one that does.
There isn't any solution that I can see after puzzling over the image for a while. The only ones that would work would be splitting them by two tiles instead of just one and using a bypass track to get around it, but that would mean you're going up to 2 tiles of width anyway, rendering the issue moot.
The only workable idea that I can come up with is to set them as two different stations and use the "Non-stop" flag for the two different trains (train A to get it to ignore the first station, train B to get it to ignore the second station) but then if your trains happen to get out-of-order you're pretty much up crap creek.
I guess the only real answer is that there is no answer.
There are some tricks for elongated stations, though no designs I know of are actually only a single-tile wide. Here's a basic implementation of the principles with a 4-tile wide double-direction design.
You can halve this design to get a 2-tile single-direction station.
Instead of 2 stations side my side or 1 in front of another station you can split them up by more tiles. Since industries rarely (or never) are adjacent to each other it should be possible to build a station on each side of the industry linking them with the CTRL placement you did with Parnpool west. Like the setup in the picture you attached, but all grouped as one station. One or two tracks on the left and one or two tracks on the right, scenario's where this cannot be done are extremely rare. There is always the bulldoze tool for cities too :)
I'm not sure what the max size of a station is by default but i never changed it and rarely get that message and i usually build huge hub stations on cramped spaces.
You really should not have any problem putting stations side by side with space, towns or industries in between them. If you do i'd like to see a screenshot of that scenario and i can probably help you out with the placement.
Ctrlwhile building a new station (or airport, or bus station, etc) you have the option to combine it with other local stations (the max spread is an advanced setting you can tweak) - this is the equivalent of building the new station adjacent to the other, creating a single unified station.