Weighted regression with no variance within cluster

Question

Given a clustered data set with no variation within a cluster, shouldn't a regression weighted with the inverse cluster size give the same results as a regression with only one observation per cluster?

Here is sample data (5 clusters with same observations per cluster) with sample code in R:

data = as.data.frame(cbind(
  c(1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 3),
  c("A", "A", "A", "B", "B", "B", "B", "C", "C", "C", "C", "C", "D", "E", "E"),
  c(1/3, 1/3, 1/3, 1/4, 1/4, 1/4, 1/4, 1/5, 1/5, 1/5, 1/5, 1/5, 1, 1/2, 1/2),
  c(2, 2, 2, 1, 1, 1, 1, 3, 3, 3, 3, 3, 1, 2, 2),
  c(4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 3, 3, 3),
  c(1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0)
))
colnames(data) <- c("y", "cl", "weights", "x1", "x2", "firsts")
data$y = as.numeric(data$y)
data$x1 = as.numeric(data$x1)
data$x2 = as.numeric(data$x2)
data$weights = as.numeric(data$weights)
data$firsts = as.numeric(data$firsts)

# Regression with one observation per cluster
summary(lm(y ~ x1 + x2, data = data[data$firsts == 1, ]))
#> 
#> Call:
#> lm(formula = y ~ x1 + x2, data = data[data$firsts == 1, ])
#> 
#> Residuals:
#>       1       4       8      13      14 
#> -0.6000 -0.4667  0.1333  0.6000  0.3333 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept)   3.3333     1.5202   2.193    0.160
#> x1            1.2667     0.7055   1.795    0.214
#> x2           -1.0667     0.7055  -1.512    0.270
#> 
#> Residual standard error: 0.7303 on 2 degrees of freedom
#> Multiple R-squared:  0.619,  Adjusted R-squared:  0.2381 
#> F-statistic: 1.625 on 2 and 2 DF,  p-value: 0.381

# Regression with weights equal to inverse cluster size
summary(lm(y ~ x1 + x2, data = data, weights = weights))
#> 
#> Call:
#> lm(formula = y ~ x1 + x2, data = data, weights = weights)
#> 
#> Weighted Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -0.34641 -0.23333  0.05963  0.05963  0.60000 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   3.3333     0.6206   5.371 0.000168 ***
#> x1            1.2667     0.2880   4.398 0.000869 ***
#> x2           -1.0667     0.2880  -3.703 0.003018 ** 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.2981 on 12 degrees of freedom
#> Multiple R-squared:  0.619,  Adjusted R-squared:  0.5556 
#> F-statistic:  9.75 on 2 and 12 DF,  p-value: 0.003056

^{Created on 2023-09-14 with reprex v2.0.2}

The coefficients are the same but not the standard errors. Why?

Side note: The standard errors also don't get similar if I use cluster standard errors.