Sometimes, two pokémon’s personalities aren’t quite compatible enough to promote a healthy relationship between them, even if they share elements and are otherwise extremely sweet and kind.
You’re on the right track with taking it slowly and giving both enough space to explore on their own and grow accustomed to their environment. If this has been going on for months, you may wish to consider taking the next step, which is a supervised, formal introduction (rather than the perhaps unsupervised chance introduction your persian has just had). Set up a time to meet with your neighbors and bring along both your pokémon. Your neighbor should leash their umbreon if they have a leash, but if not, it will be important for the both of you to keep your poké balls close by. Be sure to bring along treats or water for your persian so it will be easier to tell when he’s relaxed. Have the umbreon and your persian stay in the same room until your neighbor’s umbreon relaxes and begins to ignore your persian and until your persian seems calm and accepts a snack or water. Continue with this type of meeting until your persian understands that your neighbor’s umbreon is not a threat. Then, try an unsupervised interaction (in which you both leave your pokémon in a room) to be sure they can get along well.
If, however, your persian still seems tense during the supervised interactions, continue to keep both pokémon separate, even when you’re not at home. Try a supervised interaction again after a few days. If your persian never feels comfortable with your neighbor’s umbreon, then it’s likely that his personality just isn’t compatible with hers, and this is unfortunate but perfectly normal. That scenario will simply require you to keep the two separate indefinitely.
Of course, his avoidance of the stairs might also stem from other factors. If, for example, your neighbor’s umbreon has marked the top of the stairs (or the stairs themselves) as her territory, then your persian may be avoiding them for that reason alone. There are products on the market that you may be able to use to remove those scent markers, but it will be up to your neighbor to train their umbreon to not mark public spaces.
Alternatively, the problem may be the stairs themselves as well, so be sure to test their integrity and make sure there’s no chance that there are exposed nails or splintering boards.
Best of luck!