To me it sound like she's running scales & using rhythm & note choice to keep it interesting - blue notes, passing tones, overall tonal shape (defining the tonal centers, tension & release points, etc.), intervalic leaps & rhythm.
It's not especially melodic (not something you'd be humming for the rest of the day), so I don't think it's really a "transcribe lots of lines" thing. I also don't think it was especially in line with the "chords" being laid down underneath, so it's not a "learn to play the changes" kind of thing.
It is a lot of "I hit every note before, how do I mix it up? Let's try ascending by thirds. Now let's try different launching & landing points. Now let's try lower on the neck. Now some chords. Now using fewer notes. Now something melodic. Now vary the rhythm." etc. But underlying all that is a very, very strong knowledge of the fretboard.
The #1 thing the OP can do to sound like that is to get a real strong knowledge of the fretboard - learning scales in lots of positions so that if you're playing in C up at the 15th fret, you can decide to start down by the 7th to vary things w/o getting confused.
I'm not saying it's strictly modal. What I am saying is that a strong knowledge of the fretboard is a prerequisite - learn major & minor scale shapes & learn where you can add or remove notes to give it character. Learning modes is a good jumping off point to start getting an a) knowledge of the fretboard and b) a feel for where you can change your note choices to give the solo character.
From there, the OP could start experimenting stylistically with rhythm, note choice, intervals, etc. to bring his personality to the solo.
I did, admittedly, assume that the OP would "get" the rhythmic feel eventually, but absolutely that should be practiced, I probably shouldn't have assumed he would "get" it if he hasn't been listening to or practicing that style of music.
With the click, play sixteenths, but not straight sixteenths.
Eighths:
one and two and three and four and
Sixteenths:
one ee and ah two ee and ah three ee and ah four ee and ah
This is just a way to subdivide the rhythm. Kick drum is on one & three, snare drum is on two and four.
With a click start practicing the offbeats, varying which offbeat you emphasize each time. One day practice "AND one and two and three and four AND one and two..." Another day practice "one EE and a two ee and ah three EE and ah four EE and ah." It's a Karate Kid sort of exercise where you do it until it gets ingrained and then it just starts coming out of you naturally.
Also play a jazz bass with the bridge pickup solo'd or emphasized or at least play down by the bridge. That very sharp attack lends itself to that style of playing very well. A music man bass would lend itself to a similar, but not quite the same, style of play.
FWIW, what I think Marcus was doing was much more fluid and impressive whereas what she was doing was more "kanoodling."