Football and basketball at the D-I level, the so called money sports, generally play with different rules than the rest. Lots of athletes want to play two sports: if you want to play football, your scholarship must be for football. The NCAA rules bar players with non-revenue scholarships from playing for a revenue team – probably to keep big schools from circumventing the scholarship restrictions by giving football players cross country scholarships, or whatever. The traditional powers used to give 150 scholarships in the old days – the field just wasn’t level for MAC and Sun Belt schools. ( Chicago State has the reverse problem: it has such financial problems that it makes the basketball players run track and play baseball – since the NCAA requires D-I schools to field 14 different sports, they save money by making the major sports athletes double up. The Mid-Con (now Summit League) basically kicked them out because they were non-competitive. )
However, in the somewhat related, non-revenue sports, it is not uncommon for athletes to have partial scholarships in each sport. The most common example is athletes who have partial scholarships for cross country, indoor track, and outdoor track.