So I'm currently reading almost half of the novel and so far these are his shadows (with name)
1. Igris/Igrit
2. Tank
3. Iron (my favorite...coz he's so dumb, simple and cute)
4. Fangs
5. Kaiser/Kaisel
6. Ber/Beru
A little spoiler:
His shadows are capable of talking. So far in the chapters I read Ber is only the one talking. Jinwoo's theory is that they can talk if they attain a certain "level or rank".
Here https://www.readlightnovel.org/solo-leveling/chapter-149
But starting from that chapter the translations are not good. The previous chapters are translated very well.








So base on the explanation he's in love/interested with the alpha that's why his heat is coming whenever they're together but it will be more reasonable if it's happening because they're fated pair.
For Doyun so far, Taehan should be like The One [tm], lol. Like, it wouldn't make sense if it wasn't like that if in all his life, he's the only one his hormones reacted to...