It probably depends on the type of multivariate technique you use, but to answer some of your questions:
When we run multiple linear regression models for different dependent variables, each model will have the weights that corresponding to that specific dependent variable.
Correct. Though remember for each main effect the coefficient represents a partial effect (it is the influence on the dependent variable after controlling for other variables in the regression) and interactions are simply the product of main effects.
And when we consider a multivariate model, the coefficients (i.e weights) for each output (dependent variable) in this model are still independently calculated (I believe) for every set of independent variables.
This isn't necessarily true. Probably the most obvious example of this not being the case is a covariance-based structural equation model (SEM). These use a covariance matrix which approximates a match between an implied model and the data itself. SEMs like these are also routinely checked for global fit, something that is fairly rare or nonexistent in the multiple linear regression landscape (at least in my area), so by definition it particularly requires looking at the relationships as a whole (global fit) rather than it's component pieces (local fit), though the latter is certainly looked at in both MLR and SEM contexts depending on who is doing the analysis.
This should generate similar values for coefficients for a given dependent variable in a multivariate model and when running an isolated linear regression model.
Not necessarily. Using SEM again as an example, once you build a latent variable into the model, its pretty much impossible to disentangle that into an equivalent MLR relationship. Path models will sometimes give you the same estimates (you can literally just build an MLR in
lavaan and get the same estimates back as using a
lm fit), so in that sense they would be functionally similar.