Can we talk about univariate analysis if regression model has two dependent variables? I guess it would be incorrect. However, maybe univariate analysis means that we just analyze Y vs X1 and then Y vs X2. Please clarify this.
First off, you seem to have dependent and independent variables the wrong way round. If your notation matches conventions, then you have one dependent variable $Y$ and two independent variables $X_1$ and $X_2$. The terms dependent variable and independent variable are widely taught early (at about age 13, as I recall) and are of long use, but they are poor terms, not least because scientists often get them confused, as here, and because dependence and independence are overloaded terms in statistical science, and indeed mathematics and science generally.
I'd recommend talking of response, outcome or output variables on the one hand and of predictor, explanatory or controlling variables on the other. If people cannot agree on which terms are best, at least there is no shortage of clearer and more evocative terms than the historic terms dependent and independent. (There are yet others, but I don't want to expand too much on a side-issue.)
On univariate and bivariate: There seems to be agreement that you can apply the term univariate to methods that need only one variable, such as calculating a mean or plotting a histogram. The fact that you often apply them repeatedly to several variables is no problem, terminologically or otherwise.
However, the extension of the term univariate to regressions of a single $Y$ on a single $X$ seems to me unfortunate and unnecessary, but I have seen it before and for all I know it is common practice in your (unnamed) field.
Better terminology here if you need it might be single-predictor regression or single $X$ regression. Note that you often don't need any such term; it can be sufficient simply to talk about regression and for it to be obvious from equation, plot or table output how many predictors you have. (Indeed, if it's not obvious, then that's really poor presentation!) In general, it is not especially good practice to conflate notation and terminology, but I trust that any statistically-minded scientist, let alone any statistician, would have no trouble understanding a term such as single $X$ regression.
Another objection to the term univariate regression is that it would seem to imply that you could call the corresponding correlation univariate too, but such usage to me would be at best odd. I have to call it perverse.
In some fields, it appears that univariable regression is an acceptable term for regressions with one predictor. Look around: if the term is common in your field, feel free to use it, but I'd recommend avoiding it otherwise, as likely to be puzzling, unless briefly defined. The term univariable is to be contrasted with multivariable. (It's my impression that these terms are most likely in medical statistics, for no obvious reason except some prominent examples.)
(For completeness I have to note that regression can be a univariate method too: feed just a response variable $Y$ to a competent regression routine and it will happily return the mean of $Y$ as the predicted response.)
When we move on to regression for two predictors $X_1$ and $X_2$, this is certainly not univariate analysis; it isn't conventionally called bivariate either, as the 2 implied in bivariate is not the number of predictors, but the number of variables in an analysis; this isn't even multivariate regression, because the many implied in multivariate is in that case the number of responses; and equally it isn't multivariate analysis even in a broader sense, as just having many variables doesn't make your analysis multivariate. (40 years ago some texts would have given a different message, but language uses can shift, even in mathematics and science.)
In contrast, multiple regression remains a very widely accepted term for regression with several predictors. Whether "several" could include "one" as a limiting or degenerate case is a small vexed question. If I encountered references to multiple regression in a paper which only ever used regression for one predictor, I would strongly urge dropping the "multiple"; otherwise the term is fairly harmless. Perhaps most important is to note that the descriptor "multiple" is in slow decline; it really isn't a big deal to have several predictors, which is why multiple needs less emphasis than it did a few decades ago.
Terminology is a mine-field.
You are talking about regression in all cases. Just count the number of predictors, one, two or whatever, and make that explicit to be clear and remain concise.