Playing JV and Varsity on the same day during district.

No it is not. It only counts if a player goes over the maximum number of games they can play in one season.

UIL Game Rules for Players are

3 Tournaments and 17 Games
2 Tournaments and 19 Games
1 Tournament and 21 Games
O Tournmants and 23 Games

As long as they do not go over these numbers it does not matter what team they play on.

A player can play as many games on one night as they chose as long as they do not go over their maximum games for the year (See rookie's post). However, they can only play one day a week (not including tournaments) From the first day of the week (Usually Monday) through the end of school for that week (usually Friday 3:30ish). For example, they can not play Monday night and Tuesday night. There are exceptions to this (rainouts for example).

In answer to your post above...it is legal.

The week day rule is based on "School Nights".......

For instance, during Spring Break, a kid can play every night. That is why teams play on Tuesday or Monday and then again on Friday. Friday is not a "School Night"

In KISD JV teams play Monday and Varsity plays on Tuesday. They all play Fridays.