Happens quite a lot. My Andorian main I believe has three more rows than he should. Which makes me dread the day the higher ranks are opened up. Personal inventory pretty much already stretches from the top of the screen to the bottom. On my Klingon, somehow, I wound up with yet one more row.
What happens is this. The added inventory space is a reward for being promoted. Specifically, it's part of the promotion quest. At least in the case of Ereth (my Fed) at some point I got stuck on one of those and kept it unfinished in my quest log as I continued leveling. (I did get the rank increase, but got stuck on the ship vendor.) Then somewhere along the line the bug got fixed, I guess, and the promotion quest got reset. And I got to redo all the ones that followed as well.
For my KDF char, though, I can't pinpoint where it happened. Didn't even notice it untill recently. Maybe it happened at a laggy point. Clicked the dialogue option that generates the reward three or four times.
The game is susceptible to these things. At one time lag let me accidentally promote a BO past my own rank.
Different ship types add to your inventory size, don't they? The description for the cruiser class ships says that they have high cargo capacity (inventory).
No, your ship has no effect on your personal inventory (i.e., the window that opens when you hit the "i" shortcut key) that I've ever heard of or observed. It is solely tied to your rank progression (bugs as described above notwithstanding, of course).