The MTA did get their engineering consultants to look at the alignment of a junction of the west Red Line and the existing subway, and they found that it was feasible, with "precision engineering" and the closure of the existing subway in that area for 6 to 8 months.
The junction would take place NORTH of Lexington Market, not south. The west line would curve slightly to the north of Franklin Street to provide a better alignment. It would be heavy rail.
Building it as a "flyover" rather than a simple junction in order to avoid a left turn conflict point was also possible, but at greater cost and disruption.
To maintain the maximum degree of subway operation during the construction, Ed and TRAC recommended building the northeast extension beyond Hopkins first, with a rail yard that would allow trains to shuttle into Charles Center from the east, and then have a temporary light rail or bus shuttle between State Center and Lex Market.
New York has various places where there are parallel subway lines close together, but not for very far.
Chicago also has parallel subways only a block apart in "the Loop" on State and Dearborn Streets, with a nasty pedestrian tunnel in between, but these subways go in totally opposite directions away from the "Loop". I can't think of any other cities that do it.