Like others have said, it seems unlikely that the tuner is picking up the wrong string (make sure you mute the other strings while tuning though). It seems like you've simply got your E string tuned slightly too high, to an F. Some tuners have selectable "bass" and "guitar" modes where the tuner identifies the string but in these modes the display would normally show 1-4 for 1st to 4th string, not the actual note you're playing.
If I remember correctly the Snark clip-on tuners don't have a "guitar" or "bass" mode. I think it only has chromatic mode, which means it will display the note closest to the one you're playing. If the display shows an E it means you're playing an E (or at least very close to an E). If the display shows an F it means the string you're playing is tuned to F, which of course means it's tuned slightly too high if you're playing the E-string. Tuners with a "bass" mode will guess which string you're playing. Chromatic mode won't do this.
Chromatic mode is in no way inferior to a "bass" mode. You won't get a more precise result with a "bass" mode, just a number instead of a note in the display. Chromatic mode is actually a lot more versatile as it will allow different tunings, like drop D or whatever. A "bass" mode might be slightly easier to understand for a beginner but you'll soon learn to use a chromatic tuner with no problems at all (or even just tune by ear).
Anyway, your tuner seems to be chromatic only. If your E string is tuned slightly too high the display will show the next note which is an F. Just tune down a bit and then tune back up to E.
Actually, that last bit is very imporant. Always tune UP to pitch. If you're too high, always go down a bit and then go back up to the correct note/pitch. This is particularly important on cheaper basses where the tuning mechanism may not be the best quality. If you tune DOWN to the correct pitch your tuning stability may suffer due to backlash in the tuning mechanism.
Having said that, clip-on tuners can sometimes be problematic with the lower bass strings, giving unstable readings. Moving the tuner to a different spot on the headstock may actually help in some cases (but this can sometimes cause tuning instability with the other strings...).
Personally I never use clip on tuners with electric basses except at home when I just want to grab something fast and don't want to bother with a cable. For gigs/studio/rehearsal I would personally never use a clip-on tuner with any electric instrument that has a jack output. I've tried lots of clip-on tuners but I've never found any of them 100% stable with bass (even the ones that are supposedly designed for bass). For what it's worth my favorite clip-on tuner is the Boss TU-10. Not because it's necessarily more precise than other clip-ons but because I like the form factor. It feels a lot sturdier than most other clip-on tuners and I've not managed to break it... yet.
For gigs/rehearals I would strongly recommend a pedal tuner. They give far more stable readings with bass than clip-ons. I've tried all the fancy stuff but I keep coming back to my old Boss TU-2 (the currently manufactured TU-3 is also great). It's built like a tank and I've had mine for 30 years. After hundreds of gigs and all kinds of abuse it's still good as new. Sure, there are more precise tuners out there but do we really need that level of precision for gigs/rehearsals? I've tried all the fancy strobe tuners but I found them all to be TOO precise. I could spend ages tuning my bass and the tuner would still tell me I was out of tune. When I'm on stage I need FAST and reliable tuning. The TU-2/TU-3 is more than adequate for gigs/rehearsals and will provide fast, easy and stable tuning. Some people will say that the buffer in the Boss pedal tuners have issues with "tone suck". I've never found this to be a problem at all. It's marginal at worst and probably more of an issue for guitar players as it mainly affects the higher frequencies. There is certainly no perceivable loss of lows. I actually find the Boss buffers to be good quality compared to many other pedals. There's a good chance that any other buffered pedals in your chain (or even your cables) will affect the tone far more than the Boss tuners.
In the studio or when adjusting intonation I prefer the Korg OT-120. When precision is needed I find the visual reference of the needle far better than even the most expensive strobe tuners, even though the strobe tuners are technically FAR more precise. Again, I find strobe tuners to be TOO precise. I would spend ages setting the intonation and the tuner would still tell me it's off. Lets face it, with the exception of true temperament guitars (the ones with curved frets) the intonation won't be perfect all along the neck no matter how perfect you set it at the 12th fret. Slightly imperfect intonation along the neck is actually a huge part of the sound of guitars/basses. Those true temperament guitars sound like synths to me. The Korg OT-120 isn't ideal for a pedal board though. It's my personal favorite tuner of all times and I use it every day at the workshop but it will never go anywhere near my pedalboard.
My point is, there are lots of different tuners that are suitable for different situations. It will also depend on personal taste (lots of people love those fancy strobe tuners that I personally can't stand). I have a feeling you may be happier with a pedal tuner. Maybe even get one with a selectable "bass" mode (like the Boss pedals I mentioned) for an easier learning curve. The only real advantage of a clip-on tuner is that you won't need a cable, which is great for fast tuning at home. In most other situations you will need a cable anyway so adding another cable for the tuner is a very minor inconvenience that will be outweighed by the FAR superior tuning stability of a pedal tuner (IMO, YMMV, etc).