How much common ground is there between applied theory on guitar vs bass guitar?

Firstly, I'm going to disagree a little with the first statement. Others will disagree with that, I'm sure. If you're going to play from dots all the time, then I'll partially agree with knowing ALL the notes. However, even that isn't necessarily necessary, as sight reading often involves reading the next note from what the last one was. Not reading a C#, playing it, reading a D, playing it, but thinking D is one fret higher than C#, etc. The former is far too slow, and takes up too much brain power.

Knowing what notes are relatively positioned to others is more important. As in, you're on an A, D string, 7th fret. Next to play is a D (4th up from A,5th down). It's either on the next higher string, same fret, or next lower string, two frets down. I'm addressing beginner/intermediate here, and speaking generally.

On to the question, of which all I've said is relevant - guitar or bass. The scales and arpeggios are a given - for any instrument! Except that a lot of playing expected from a bassist is similar patterns - ostinati, if you like, which change with each chord change. Root and 3 and 5 figure a lot,as these notes arpeggiate the chord being played. Hence the need to know where a G# or a Bb is - although knowing their addresses on E and A strings will often be enough to start the arps going. It's patterns after that.

Rhythms and note value knowledge are very important, too, as the bass is part of the rhythm section - part of a team, generally, and as such keeps the others together, on track, in time, along with the drummer. That stuff isn't so important for a lead player - although it's paramount for the rhythm guitarist in any situation.

I suggest for more insight into an answer to this specific question, look at RGT exams for electric guitar and bass. You'll see there is a lot of similarity, but more and less emphasis on some facets.

And, of course, the tuning of the lower 4 guitar strings mirrors the 4 strings on a standard bass, albeit an octave higher, so any theory relating to that phenomenon will be shared.