well - shop A isnt interested it seems in doing the piece you want
so the time discrepancies are a mute point
and you shouldnt be overly focused on time
full sleeve is gonna take you awhile to complete
if you have the dough and feel confident about your idea and the artist
go for it
Time etc is a bit of a moot point, tattooists are people just like you, we're all different! We work at different rates, and to be honest I think it's pretty impossible to say how long a 'sleeve' would take as it's very dependent on you and how well you sit. I'd say with something like you want you really need to do your research. Look at portfolios, do the artists have experience of doing similar work? Shop A sounds like they put you off because they don't have anyone capable of doing justice to the idea, but instead of pointing you in the right direction (eg. We don't have a realism artist at our shop so we send people who want that to someone we recommend in our city centre) they tried to talk you out of it.
I'll reiterate how important I think it is to research and then if you're happy with the artist you have chosen (you trust them to do got a good job) you can usually trust them not to fleece you time /money wise. Large work like that takes as long as it takes!
You're question 'what is a tattoo?' I'd guess is a response to what the first shop said. I suspect that they think the first one certainly is photo shopped, which is entirely possible. That being said, it's tattooable by the right person. They might also think, like I tend to do, that a proper 'tattoo' needs black lines in it to hold it together over time. Realism like that can blend/fade into a mush, especially if done poorly. Tattoos need structure, but that's just my opinion,
