I have used Strat and Tele pickups many times in bass builds. They sound, basically, like bass pickups. If you want a pickup leaning more to the treble side, certain strat pickups lean that way. Dimarzio have tone specs for all their pickups online. I am sure others too.
There is a misconception that there is something fundamentally different about bass and guitar pickups. There are differences about how the pole pieces line up, of course, but there is about where the differences stop - guitar and bass pickups are much more alike than most folks think they are.
Magnetic pickups (what we use on guitars and basses) share the same Physics. One of the things that brings with it is that there is no (zero) low end rolloff in magnetic pickups - a guitar pickup will have every bit as much low end as a bass pickup if put on a bass. Also, there are not "scooped" pickups - magnetic pickups are very flat at low
and low mid frequencies. There is a high mid peak - where that is frequency wise, and how high it is are the primary things that make one pickup different than the next one. Humans have preferences for where they like to hear those peaks - pianos, acoustic guitars, violins, etc. have them too, but they are done acoustically in those instruments. Above that peak, whether it is done electrically or acoustically, instruments roll off response wise - we don't like a lot of really high treble (it sounds harsh).
If someone tells you their bass has "hi fi" pickups, the implication is that they have flat response. They don't - they may have more extended treble (a higher frequency mid peak), but there is a peak and a rolloff - without the rolloff, the bass would not sound "hi-fi", it would sound awful. I have measured responses of a couple hundred different pickup types to date, and all of them have a peak (though it may be highly damped), and a rolloff.
Pickups get their characters from where their upper mid peak is, and how high it is. If a pickup sounds "mid scooped" or "bass shy", it isn't - but it may be that your ears are distracted by a lot of upper mids, and your brain is interpreting that as something other than what it really is.
