Generally speaking, the UART you're talking about is enabled as /dev/ttyS0 on the clanton-tiny distro (ulibc one) and you said you tried that one.
One thing that drew my attention though is this:
RX to RX, TX to TX
I'd expect you to rather connect it RX-to-TX, TX-to-RX, can it be as simple as that? :-)
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As far as port names are concerned, I wouldn't expect ulibc version to be different from eglibc one, but it's always worth testing out, so it's good you're going to do that.
What I mean is that you need to connect the transmitter (TX) on one end to receiver (RX) on another, if you connect receiver to receiver both ends will just be listening and with transmitter to transmitter both ends will be blindly transmitting, to put it in a very simple way :-) You can e.g. refer to the standard PC serial port cable pinouts - you'll see the TX and RX at one end connect to RX and TX respectively at the other.
I've also just looked at the tech documents for the module you're referring to on the Adafruit website and I'd also suggest you to double-check your power connection. You mentioned you're connected it to the VIN and per the Galileo docs it's providing 5V while per the Adafruit docs (I used the one for version 3 of the module) the GPS module accepts 4.3V max (so it may be a problem too). Looking at your photo, it looks like you've actually connected the module to the 3.3V pin (if so, you should be fine), but it's hard to see there. Also, out of the docs, at least as far as I understand that, you'd need to set the IOREF jumper to the 3.3V position to have 3.3V supplied. It's easy to check by simply measuring it, but I don't have the board at hand right now to do that.
EDIT: Ok, never mind - looking further I found out these voltage limitations are for the GPS module itself, while for the Breakout the wider range of 3.3-5.5V applies (here), so Adafruit not only provides breakout there but a power regulator. So disregard the above, just make sure you have some voltage there, either 3.3 or 5.
Here's the same point regarding crossing RX and TX from the Adafruit's tutorial for that GPS module (the second paragraph):