I think you have two related concepts here: #1 is how standard error (SE) relates to sampling variation, or how precisely we can estimate a population mean based on sample data; and #2 is about the difference between a standard error and a standard deviation.
Some of this confusion arises because we use similar-looking tools in different settings (explained more below).
1. Standard error and estimating the population mean
The short version here is that standard error [of the mean] is an estimate of the extent to which sampling variability might be impacting our estimate of the population mean, based on our sample data.
We often use the t-distribution to think about this hypothetical distribution of sample means, as in smaller samples this is a better option than using the Normal distribution (which tends to lead to underestimation of statistical inferences i.e. too liberal). As noted in a comment to your question, as sample size gets larger the value of $t$ from the t-distribution (which depends on sample size) converges towards the value of $Z$ from the Normal distribution.
2. Standard error and standard deviation
As above: when we're talking about standard error of the mean, we're talking about the expected/estimated variation in repeated sample means drawn from the same population.
When we're looking at standard deviation, we're considering variation in individual observations around the mean: typically variation of individual's values around the sample mean (e.g. if we're estimating mean height from a sample of students).
This can sometimes be re-scaled into a Z-score, which is the difference between a single observation and the mean divided by the standard deviation (see formula below). The Z-score is then the number of units of standard deviation a particular observation falls from the mean in that sample.
$$\frac{X_i - \bar{X}}{SD}$$
Footnote 1. Z-scores can be calculated for a given observation relative to a reference population (e.g. one real-world example is a child's body mass index [BMI] expressed as a Z-score relative to the mean and standard deviation of a reference population of children of the same age/sex)
Footnote 2. Z-statistics are sometimes calculated in hypothesis testing too, but I'll ignore that here.
