Can you fit a Cox-PH model to left censored data? Yes. Left censoring can be a subset of interval censoring, and you can fit a Cox-PH model with R's icenReg package, using the ic_sp function.
However, rather than blindly plug my own software, I will ask if you really want to fit a Cox-PH model to your data. I'm not saying you don't, but just knowing that data is left censored should not be the defining factor for fitting a Cox-PH model.
Recall that with censoring, we consider there are two processes: $T$, the true response value (represented as $T$ to denote event time in the traditional survival analysis case), which potentially is not fully observed due to censoring and $C$, the censoring process. A Cox-PH model describes the relation between $X$ (covariates) and $T$, and as such we need to be thinking about this relation when deciding to use a Cox-PH model. It just happens to be computationally really convenient to calculate when the process $C$ results in right censoring (and all the computational convenience goes out the door with interval censoring).
So before we can decide that a Cox-PH model is appropriate, we need to consider $T$'s relation with $X$. If you move outside the world of survival analysis, it becomes really difficult to interpret covariate effects: for example, what does the "hazard ratio" of salary even mean? And even if are not so concerned with interpretation, you need to ask if the effect is really going to be appropriate; does the regression relation $S(t|X, \beta) = S_o(t)^{exp(X^T \beta)}$ really seem to describe what you are seeing in your data?
In summary: yes, you can fit a Cox-PH model to left censored data. But whether you should is very dependent on the relation between $X$ and $T$.
I think that many people who use the words "multivariate regression" with Cox models really mean to say "multiple regression." (I will confess to having done that myself; it's common in the literature.) "Multiple regression" means having more than one predictor in a regression model, while "multivariate regression" is a term perhaps better reserved for situations where there is more than one outcome variable being considered together. In a Cox regression you are typically modeling just a single outcome variable, survival of some sort.
If you are preparing results for publication in a medical journal, the editors and reviewers will typically expect to see a table of single-variable relations of predictor variables to outcome (your "univariate" regressions). These single-variable relations, however, are seldom very informative due to relations among the values of the predictors and potential interactions among the predictors with respect to outcome.
These issues can be handled by Cox multiple regression, which gives you the best chance of evaluating each of the predictors with all the others taken into account, and which allows directly for testing of interactions. You have to be careful not to evaluate too many predictors together in a model, however. A useful rule of thumb is that you should limit your analysis to no more than 1 predictor per 10-20 events (recurrences or deaths in oncology) in a standard Cox multiple-regression model.
Note that there can be a true multivariate Cox regression that evaluates multiple types of outcome together (e.g., both recurrence and death times in cancer studies), or that treats multiple events on the same individual with multivariate techniques, as in standard multivariate linear regression. This paper is one often-cited reference, in case that is what you actually mean. But in my experience, I think most people in the clinical literature say "multivariate Cox regression" when they really mean "Cox multiple regression."
It would be wise to get some more direct advice from a local statistician, as there are many issues that need to be considered in building a reliable survival model. Working with an experienced practitioner can also be an efficient way to learn for yourself.
Best Answer
Abbreviated Model Descriptions
The Cox model is a survival model that cleverly models the hazard ratios through the observed ranks of the data, without needing to make an assumption of the underlying baseline distribution, but still requires the proportional hazards assumption.
The Tobit model is essentially standard linear regression, except that it can also handle censored data. The assumed distribution is then normal.
Pros and Cons
Cox Model:
Pro: Don't need to make assumption about baseline distribution. This is very important for survival analysis: time-to-event data tends to be very not normal, often with extremely heavy right tails. Additionally, by only considering the rank of the data, you have a model that is more robust to the expected outliers.
Cons: Can be very difficult to interpret coefficient effects.
Tobit Model:
Pro: Simple extension of a model most analysts are already familiar with to allow for censoring, i.e. if all your data were observed and appropriate for linear regression (with one caveat mentioned in Cons section), then it would be appropriate to use a Tobit model.
Cons: Requires the assumption of linear effects and gaussian errors. In some applications, this is totally appropriate, but time-to-event data (i.e. survival analysis) rarely fits that criteria. Also, it's worth noting that the Tobit model is more sensitive to the normality assumption than vanilla linear regression.