The Ferrari 248 was so-called because it was the first year of the switch to 2.4 litre V8 engines. Previous Ferraris followed this template, e.g. the 312 series (312, 312B, 312B2, 312T, 312T2, 312T3, 312T4, 312T5 etc.) for 3-litre V12s. The F310 of 1996 followed the same "logic".
The Midland M16 was a continuation of the previous Jordan numbering scheme, which had got up to EJ15.
Many teams, including Benetton, Renault, Jordan, Minardi and Toyota use numbers based on the year of competition. Perhaps their inspiration was Max Mosley's old team March, who designated cars according to year and series, e.g. The March 701 was a 1970 F1 car; the March 702 was a 1970 F2 car etc...
And, in case anyone doesn't know, McLaren originally numbered their cars simply "M", ending with the M30 in 1980. When Marlboro organised for Ron Dennis to take over control in that year, the new designations reflected the merger between McLaren and his team - Project Four - hence MP4.