Specifying Taxonomy Term Filter in View from Page Editor
There are two parts to this question - one is to have creators of nodes to choose a Taxonomy term, the other is to have a Views Display show up on Node View based on the Taxonomy term set inside the node. 

Taxonomy term reference

As you've already worked out, the easiest way to allow for users to influence what content list to display on their nodes is to allow them to choose a Taxonomy term. 

You can do that by creating an Entity reference field referencing Taxonomy terms. The way you worded your question suggests you will only allow one term to be selected (limit number of values to 1). 

When the node is saved, the information about the chosen Taxonomy term will be displayed when viewing the node. If you don't want this to be visible, if you only want this field to serve as a way to influence Views in the following steps, you can hide the field from showing up on node view by changing Manage Display settings. The info from the field will still be available for Views to read "on the sly". 

Views Block listing content sensitive to the Taxonomy of the page they're on

Creating Views that display related content is probably the most common request in Views. With such a Views Block you don't have to have any additional interface for node creators to choose what to display on their nodes. Simply having a Taxonomy term reference is enough (already covered above).

First you create a Views Block listing your desired Content (set up all the regular Filters and Fields). 

Then add a Contextual Filter of type Content: Has taxonomy term ID, and configure it to Load filter from node page (NOT taxonomy term page). Take a look at screenshots from this answer: display lists of related content that share the same Taxonomy term. 

In the end place the Block in a region, and configure the block settings (to display only on node page where you want). 

When your content creators save a new node, that Block will automagically show only Teasers of content tagged with the term they chose when creating the node. 

If the content type of the created node is the same as the content type of teasers in the Block, the newly created node will also be displayed on the Teaser list. If you don't want the node you're currently viewing to be listed there you can exclude it with an additional Contextual Filter Content: ID (Provide default value: Content Id from URL, More (at the bottom): Exclude).