Is there a way to copy text style(font, weight, etc) in Photoshop?
Yes, there is. Ctrl + click (with + shift or shift alone, if there are many) all text layers in layers dialog to select them, then in character settings window, just set italic, font-family, weight, etc. For layer style, has been explained above.
Character Style and Paragraph Styles are designed for just this.
Much like layer styles, you simply format a text layer how you'd like it then click the little "new" icon at the bottom of one of the style panels. This will create a new style. When you then wish to format text to look like that style, simply highlight the text layer and click the style in the Character Style or Paragraph Style Panel.
Character styles will alter just the text - font, face, size, etc. Paragraph Styles will alter paragraph options - alignment, spacing, etc.
Layer style? Or actually text style?
Right click on the layer to copy the layer style. Right click again to paste on the new layer.
For the text formatting itself, duplicating the text layer is the easiest option.
Duplicate the text layer? That's what I wind up doing.
If you select a text layer with the text tool it will copy the text style. But the text size isn't copied across if you use transform tool. If you use the font size selector it works as expected.
press and hold alt and drag the text need to be copied. All layer styles will also copied.