Automatic post excerpts are trimmed to a specific character length, which means it’s possible for HTML tags to be broken. You could end up with an open tag not being closed, for example.
To avoid this WordPress strips HTML tags from the content before trimming it for use as an excerpt. The are other reasons WordPress might want to do this. For example, it could be undesirable for an image to be included in what’s supposed to be a short excerpt from the content.
If you want to use HTML in excerpts you need to manually write an excerpt into the “Excerpt” box on the post edit screen or, if your theme supports it, manually define a cut-off point in the content by using the Read More button in the editor. It looks like a dotted line splitting a box in two.
Some themes might not use excerpts at all and just show the full content, unless you have used the Read More button, in which case all the HTML will be preserved.
Ultimately the handling of excerpts comes down to the theme, since the theme decides which of the method/s of excerpting posts WordPress provides that it wants to use.