I can, say, highlight text within blockquote:
> This is a blockquote and ==this part is highlighted.==
But highlighting does not work inside, say,
<blockquote>
This is a blockquote and ==this part is highlighted.==
</blockquote>
What is the way to make it work?
It is because you can use either Markdown or HTML, not a mix of both as in your second example.
Your options would be:
> This is a blockquote and ==this part is highlighted.==
or
<blockquote>
This is a blockquote and <mark>this part is highlighted.</mark>
</blockquote>
Somebdy may correct my but to my knowledge, it can't be simply done with any flavor of markdown. But a little tinkering with html might help, resulting in something like this:
<pre> This is a blockquote and ==<b>this part is highlighted.</b>== </pre>
This will look like this:
This is a blockquote and ==this part is highlighted.==
If this suffices, just reorder this a bit (add a new line after the opening pre). 
My statement above was not quite right, you can use HTML in Markdown, but not Markdown in HTML.
So
> This is a blockquote and <mark>this part is highlighted.</mark>
will also work in Joplin.
This was not the OP's specific question but has just been added for correctness.