Obviously "same" is relative on two different things. Shall we say the GOAL is to have them sound as much alike as possible.
I can say that it it common on cheapo basses for the open E to ring out and the E on the B string to be a dull thud. NOT the '"goal"! But we are talking fretted basses here which means a finger fretting a note is not supposed to be damping it. And on a decent bass it won't. And string difference again depends on the quality of the strings you are using (and scale length) even if one can argue that they are never going to be absolutely identical because of gauge differences.
I have a L2500 with TI Superalloys. The b string is one of the best I tried so far, and these strigs are simply awsome. I compared my L2500 (wich is not in the same league) with Adamovic and Ken Smith that I tried, and is at least at the same level (speaking of tone here, the feel and the constructions on these Adamovic and Smith´s are superior).
My L2500 is modified: Ken´s bridge mod, changed preamp chip with its burr brown counterpart, and eliminated a condenser on the preamp board because it was muffling the sound. These changes opened up the tone in general and specially above the fifth fret, and the b string response is the same as the other 4 or maybe a tad stronger.
On recordings I hear any difference when playing the b in comparison with the other 4 in terms of tone. The volume is stronger on the B, that´s the only diference I hear.
And one other thing about the fretted E on the B and the opened E. On bass amps I can barely hear diferences but I do my tests on Genelec monitors. That´s why I hear all these diferences
