its a very complicated matchup that is vastly different every time depending on each champions runes, masteries and item setup.
shyvanas setup aside, renekton can ONLY win this matchup if he builds basically full offensie bruiser with hydra, BT and LW. ´
once tiamat is complete, you are able to outpush her if she comes to trade with you in a minion wave due to your healing from empowered Q and tiamats splash. tho you can probably never kill her when she goes defense, except if she really tries hard to trade, you CAN duel her for the most part of the game and control her laning, jungling and roaming while denying her some CS.
if shyvana builds damage and goes with the smite but not for cinderhulk but for warriors or even plays without smite entirely, its a matchup where one will kill the other and snowball the lane, but renekton with damage has the upper hand due to him having a stun, which allows him to do quick trades without much counteraggression from shyvana.
tho if you trade with her when she builds damage, you can NEVER engage with E, do your combo and disengage, as she will R after you and kill you in 2 seconds because both of you built damage. you can only walk up to her, combo her (dont forget auto attacks during your combo) and then double E away through a minion wave to wait for your cooldowns.
basically, if played correctly, renekton WILL win this lane if he builds damage, which is at the same time the superior mid-lategame build, so you are sure to have more impact on the game than her.
if shyvana runs smite, its your job to follow her during jungle attemps and prevent her from stealing your junglers camps and buffs. if she has no smite, just beat her in lane and do some good groups with your team. it is recommended to run ignite on renekton.