PowerShell 7.1, How do I Get Aliases from a List of Functions Names

this command returns an error.

You can simply ignore these errors, using the common -ErrorAction parameter, which simplifies your approach:

Get-Command mts_* | ForEach-Object { Get-Alias -ErrorAction Ignore -Definition $_ }

Caveat: This won't find aliases of commands from auto-loading modules that haven't yet been loaded (imported) into the current session; note that such modules are implicitly loaded the first time you call any of their non-alias commands.
Similarly, Get-Alias only reports aliases from currently loaded (imported) modules, so the caveat also applies to Scepticalist's answer, which additionally has the disadvantage of having to call Get-Alias to retrieve and filter all aliases for every input command name.

Note that Get-Alias's output formatting includes both the alias name and the definition (target command), so that you'll see something like:

CommandType     Name                                               Version    Source
-----------     ----                                               -------    ------
Alias           foo -> mts_foo
Alias           bar -> mts_bar

If instead you want a two-column display limited to the alias name and its definition, you can use Select-Object:

Get-Command mts_* |
  ForEach-Object { Get-Alias -ErrorAction Ignore -Definition $_ } |
    Select-Object Name, Definition

The advantage of using Select-Object is that it outputs objects suitable for later programmatic processing.

You'll get something like:

Name Definition
---- ----------
foo  mts_foo
bar  mts_date

However, in order to control the display output you can instead use Format-Table, which gives you more control over the formatting - but prevents subsequent processing of the output, given that Format-* cmdlets output formatting instructions rather than the original data.

For instance, the following gets closer to your original output format:

Get-Command mts_* |
  ForEach-Object { Get-Alias -ErrorAction Ignore -Definition $_ } |
    Format-Table  Name, { '->' },  Definition -HideTableHeaders

The { '->' } script block is a (simple) example of a calculated property, which are typically hashtable-based (@{ ... }), and in the case of Format-Table even allow you to control the column width via a width entry.

You'll get something like:

foo  ->     mts_foo
bar  ->     mts_bar