Is there a way to own a horse but have it live elsewhere?

@simsplayer818 @MighyMsHolmes I did a similar thing to what you mentioned, simsplayer818. I moved the 'dad' horse in with a friend, and I can then go to my Sim's 'relationships' list and click the 'dad' horse there and see the options below... so I then choose to "Invite to Hangout at Current Lot". He will then come over and hang out on the lot for the day. It's not perfect, but it works. In my case I moved the horse into another of my households that was not a family member and that was not placed on a lot with a barn, etc. for the horse (see screenshot)... so in my case going to the other Sim's lot to interact with him would not be an option. I think using simsplayer818's idea of moving the horse in with a family member with a barn you can use to also visit them is a great idea, even if you use both methods to interact with them and continue the relationship. In my case, my biggest concern was my goal of keeping the relationship up with him and the 'mom' and 'baby', so I wanted him on the lot for those purposes. 

I was thinking that with your goals in mind, using the workaround that @simsplayer818 mentioned is the most primary option, but perhaps keep in mind that you may have some fun playing with inviting the horse to hang out with you sometimes also, even if you have no barn on your lt. Perhaps your Sim could move into a lot next to or near to a lot with training and riding facilities, train/ride where you want, then the horse can go back home to meet it's needs. I think there could be a number of combo methods of doing this that you could have some fun trying out and seeing what works best for you and your goals. Have fun!