Least-squares estimation on a general SUR model is not the same as Pooled OLS on a Panel Data model (at least given the meaning with which these model labels are usually endowed).
Both models are "System of Equations" models, but the SUR model sprung from the observation that in many cases, the disturbance vector from each regression equation is correlated with the disturbance vectors of the other equations. So the general SUR model does not impose the restriction that the unknown coefficients under estimation are identical across equations, as is the case with the usual Panel Data setup and Pooled OLS. Also, in benchmark Panel Data models, errors are assumed independent across equations.
In symbols, the general SUR model is ($N$ is number of equations/cross sectional samples, $T$ the number of available observations for each cross section)
$$y_{it} = X_{it}\beta_i + u_{it},\;\; i=1,...,N,\;\; t=1,...,T,\;\; E[u_{it}u_{jt}]\neq 0$$
while for a Panel data model and Pooled OLS we usually have
$$y_{it} = X_{it}\beta + u_{it},\;\; i=1,...,N\;\; t=1,...,T,\;\; E[u_{it}u_{jt}]= 0$$
Note the two differences.
A usual example of SUR: you have data on a few big firms from the same national economy related to their financials (e.g. sales, profits, market share, credit lines etc), and their investment spending (dependent variable), and also to certain macroeconomic indicators for this national economy. You can reasonably argue that
a) at least some of the coefficients of an explanatory variable are not the same across equations-since how, say, profitability affects investment spending may depend on the long-term strategy of each company, which is formulated by different decision makers, etc
and
b) that at least partly some shocks/disturbances/other factors that are included in the "error term" in each regression, are common to all equations.
So you are led to the SUR model. Cross-equation restrictions on SUR models may arise -but again, they are not usually of the kind imposed in the Panel Data literature (ideally we should have one unified "Systems of equations" regression estimation theory with all the various models as special cases -but we don't).
The attempted benefit from a SUR specification is gains in estimator efficiency, and a usual estimation method is (Feasible) Generalized Least Squares.
The specific paper you mention suffers from a serious deficiency -the authors do not write down the theoretical equations describing their model, neither do they describe clearly their data set. In the beginning I thought that indeed, they had essentially implemented Pooled OLS -but this is not so, they have run SUR after all, as follows:
Their sample does not have a time-dimension (so the index "$T$" here does not represent time). Their dependent variables are "Infant mortality" , "Child mortality", "Child malnutrition" indices. So they have three equations , $N=3$. For each of the 43 countries they have $5$ such values for each dependent variable (sub sample averages), one for each asset quantile. So the $T$ dimension for each equation should have, as they say, $5 \times 43 = 215$ observations for the dependent variables (in practice they have only $T=175$ due to missing values of the dependent variables).
For each equation $i$, and for each observation $t$, they have as regressors a constant, dummies indicating the asset quantile from which the value of the dependent variable came, etc, without indicating that a specific value of the dependent variable has come from a specific country (but the regressor values of course are related to the country that the value of the dependent variable came from). So no idiosyncratic "country-specific" effect, as such (since in any case there exist regressors that refelct various aspects of each country).
So in, say, Table 3 of the paper, the first three columns (labaled 1,2,3) is a SUR system, while the next three columns (labeled 4,5,6) is another SUR system with some changes in the regressors. So, say, the coefficient on the regressor "GDP per capita" is indeed different for the three equations of the first SUR system.
The data series of the regressors for the three equations in each SUR system is identical except for the dummies indicating "asset quantile". So here SUR estimation is not numerically equivalent to equation-per-equation OLS.
The commentator "Fixed Effects" suggestion proposes a different approach: Specify $45$ cross sections/equations, each cross section a dependent variable from a specific country. For each cross-section, take the other dimension $T=5$ to be the $5$ asset quantiles (so no dummy variables for them here). (or vice versa). Etc. This would appear to suggest that we should specify three distinct FE models for each of the three dependent variables, i.e. not estimate jointly the three dependent variables... which again brings us back to the need to validate the SUR specification, by showing that indeed there appears to be correlation between the disturbance vector across equations -something that the authors appear that they haven't done/presented in the paper.
Best Answer
Adding the year dummies to your regression is unlikely to solve the problem of unobserved fixed effects. You should definitely include them both in OLS and fixed effects regressions to account for annual fluctuations in your dependent variable that were not due to any of your explanatory variables.
Only the fixed effects estimator (or first differencing) eliminate all unobserved fixed effects. As you said, also the observed time-invariant variables (like gender or similar) get dropped but this is not a problem unless you are interested in actually estimating their coefficients. Their contribution to the variation in the dependent variable just gets absorbed in the overall individual fixed effect. If you are interested in estimating the coefficient of these time-invariant variables, see here.
Note that panel data models need a correction of the standard errors for serial correlation (e.g. by clustering on the individual's ID variable). This might be the reason why your OLS standard errors are so small.
In order to decide whether you should use OLS or fixed effects you can use the Hausman test. The test compares the consistent but inefficient estimator (fixed effects) to the potentially inconsistent but efficient estimator (OLS). If the unobserved fixed effects do not bias your results, OLS and the fixed effects estimator should not differ significantly from each other. Then you can use OLS, otherwise you are better off with the fixed effects estimator because your point estimates will be correct but trading off some of the efficiency, i.e. the standard errors will be bigger.