Great question OP, even though it's been a debated topic elsewhere and in conversation, I do think it's become more relevant to explore now that we live in a world where Shenmue 3 is actively in development and Yakuza has yielded a lot of success and potential growth as a franchise for SEGA.
When I first heard about Yakuza (fyi: I was young, like in secondary school) it seemed like SEGA were selling out, with all the magazines I used to read mentioning "GTA of the East" and seeing the images/vibe/cover image everything seemed as if it was the antithesis of what Shenmue was about. It looked like a reactionary game (even though it wasn't and was infact the first mature-rated game SEGA ever made) but the focus on japanese gang culture was a huge put-off.
I remember playing Yak3 at a friend's house (using their save file xD) seeing the openworld and trying for the very first time to approach the game like shenmue and it wasn't working out at all: I'd try to talk to NPC's and nothing happened, no in-game time and dynamic weather etc (which is one of the most pivotal/crucial shenmue elements to me, those elements make you approach the game very differently full-stop) and the lack of VO and limited selection of minigames made the game feel very much inferior to shenmue.
A couple of years or so later I played 1/3rd of Yakuza 1
and in that third of gameplay there were aspects I could appreciate but the combat was unrewarding and repetitive. I have to play the game acknowledging that it was a step down to shenmue, despite being a game that came out many years later. I enjoyed some of the shenmue-esque aspects of the game, but that English VO was atrocious (and not endearing in a way like shenmue or resi 1, just baaaad) and I couldn't handle it anymore. I thought from then Yakuza just wasn't for me.
Then, Shenmue 3 was announced. I had my hands on a PS3 at the time and decided to give Yakuza ANOTHER GO after a GT/EasyAllies video with Huber and Ben really pushing it any saying "look, don't be intimidated by the amount of games, just jump into Yak3 or 4 and watch the flashback cutscenes from the other games and enjoy" so I did that and jumped into Yak3. I took it slow (beat the game in around 9 months) I thought that the game was okay, I liked that it did some unique/unexpected/mundane stuff that was very shenmue-esque but the gameplay felt very limited (finding out later that sega did a poor/budget job of localising it and stripped out basically a third of the game). The story was compelling and there were definitely some cool moments that occur during normal exploration which I liked a lot but pacing out that game was the best thing I could do to avoid feeling burnt out on the game. The combat was much better and versatile compared to the others and I liked the new camera perspective.
I had Yak4 and continued on, and it was here that I felt hooked to the franchise. It felt like a huge step up from 3 in every regard, and I liked the addition of multiple protagonists with different fighting styles helping the game's pace. the story while a bit convoluted at times was really engaging and the side stories were very cool. Yak 5 was the same level of experience but beefier and more polished.
On thing's for sure: If SEGA was around and still making hardware, Yakuza wouldn't exist, whether Shenmue 3, 4, 5 etc ever came out or not, they would have continued to focus on games made for the team demographic and under (like nintendo) for the most part as their mindset was that gaming was for them and it took extreme humility for them to step away from that...
I definitely felt both extremes about yakuza in my life. I ain't no psychologist, but I do wonder to this day if Shenmue 3's announcement made me a little more accepting of the Yakuza series and seeing it for itself without having to scratch every Shenmue itch. Overall Yakuza definitely fulfil's some of those shenmue desires, and has continued deliver that through iteration of each game. I'm looking forward to seeing it grow bigger and better and striving for that Shenmue 1 and 2 originally represented with a certain level of ambition with immersion, but in a different and practical way (as their focus won't be on having real-time clocks/npc interaction and simulation, but evolving the formula of what exists).
I'm glad shenmue exists and is happening, as over the years we've only seen aspects of it's mechanics/vibe between several games (heavy rain, l.a. noire, sleeping dogs etc). To me the passage of time and character/personality the openworld has with the music and focus/interaction on the mundane and minutiae is unprecedented. That's what i look forward to with shenmue 3.
...sorry for the wall of text :(