Every bass guitar I own has its own character, yes. And they do make me come up with different lines/riffs...
But some have a wider palette of usable sounds then others.
Besides the tonal aspect there's different types too. I have a fretless 5 and a fretted 8 with octave strings. Of course they make me play differently.
Also, some basses are more responsive to your playing then others. My Warwick is very forgiving because it seems to have natural compression going on but this also means it kinda always sounds the same, I blame the Ovangkol it is made of. Anything Maple or Ash will generally respond much more to how soft or hard I play and "speak" more easily.
So I have always thought I needed a number of different basses to keep being inspired and able to play different styles.
HOWEVER since a couple of weeks I own a used market Fodera NYC Empire 5 (which is basically a Jazz with a really good preamp and one I could afford). This is the first bass that seems to be able to pull off just about anything. The mind got boggled good by it. It is articulate yet gels properly in the mix, growls really aggressively when I dig in, becomes piano like mellow with a light touch, gives me that Marcus Miller spank when I slap it. Has enough mids going on to punch through mixes well. I am completely in love with this bass so now thinking of actually selling the others (Warwick Dolphin, Zon Sonus, Zon Legacy, Ric v63) as I do not think I will be playing them much anymore. Have found a holy grail? Maybe. I am not making any hasty decisions though and will hang on to them till after that "NBD" rush has calmed down a bit.
So my answer is this:
A lot of basses affect your playing because they simply don't do the other stuff well.
BUT there are basses out there that can cover a LOT of ground in the total bass landscape and keep you coming up with new sounds and playing styles.