A full backing track has SO much more harmonic power, with bass and harmony instruments, it rules and bulldozes over your feeble little solo notes. Mode is not just a set of note names, because one of the notes is the King, home note, harmonic center of universe, the tonic. C Ionian and D Dorian have all the same notes, but a different center note.
Mode = TONIC + intervals around it
The feeling of a mode comes from the combination of all pitches and sounds the listener hears, the "total music", which is also affected by the rhythmic placement of notes in relation to a pulse/meter. Even unpitched noises or clicks can affect the harmonic perception and weight of pitched notes, if the clicks create a rhythmic pulse grid, suggesting where the "one" is rhythmically. If your rhythmic centerpoint moves, it might move the tonal centerpoint as well! Beginners sometimes incorrectly think that to establish a modal feeling it's enough to select the right set of notes, but that's not true at all. HOW and WHEN you play the notes is extremely important, and so is what OTHER instruments are playing.
The lowest note i.e. bass is a very powerful factor in tonic-setting. If the bass plays C all the time, it will almost certainly make C the tonic note, and any modal feeling that's built by other notes above the C will be C something.
If the backing track is playing/vamping on C major, there is no way for a soloist to make it sound like D Dorian, because D Dorian would require D to be the tonic note. No can do. BUT if the backing track is playing C major and none of the backing instruments play an F note anywhere, then the soloist CAN make it sound like for example C Lydian, by playing the characteristic F# note. Or if the backing track doesn't play B, the soloist can make it C Mixolydian by playing the characteristic Bb instead of B.
If the backing track is playing/vamping a major chord, the soloist cannot make it sound like any minor mode. Minor modes are: Dorian, Phrygian, Aeolian, Locrian.
If the backing track is playing (vamping) a minor chord, the soloist cannot make it sound like any major mode. Major modes are: Ionian, Lydian, Mixolydian.
If the backing track only has a C bass note and nothing else, then you have total modal power and can make it sound like any C mode. But you cannot make it sound like e.g. D anything.
(The only real exception I can think of to push the tonic away from a fixed bass note is to utilize perfect fourths/fifths. If the bass plays C, it will set the center to C, EXCEPT maybe if you play a heavy F note and things from e.g. F major or F minor - then maybe you can convince the listener that it's just a very long held F/C or Fm/C chord, waiting to resolve home to F or Fm.)